Section1.1

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Last update: 30/06/04

2004

Author(s): Heinz Hanßmann, Naomi Ehrich Leonard, Troy R. Smith
Title: Symmetry and Reduction for Coordinated Rigid Body Networks
Preprint: RWTH Aachen
YEAR: 2004
(MASIE)Subsection : Related publication
Abstract: Motivated by interest in the collective behavior of autonomous agents, we study networks of rigid bodies and the problem of coordinated orientation and position across the group. Our main result is the reduction of the networked system in the case that individuals are coupled by control inputs that depend only on relative configuration. We use reduction theory based on semi-direct products; this yields flat Poisson spaces which enable efficient formulation of control laws. In the second part of the paper, we apply the reduction results to particular choices of kinetic energy and prove stability of coordinated behaviors.

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Author(s): Arturo Ramos
Title: Poisson structures for reduced non-holonomic systems
Journal: J. Phys. A: Math. Gen 37 (2004), 4821-4842.
YEAR: 2004
(MASIE)Subsection : 1.5
Abstract: Borisov, Mamaev and Kilin have recently found certain Poisson structures with respect to which the reduced and rescaled systems of certain non-holonomic problems, involving rolling bodies without slipping, become Hamiltonian, the Hamiltonian function being the reduced energy. We study further the algebraic origin of these Poisson structures, showing that they are of rank two and therefore the mentioned rescaling is not necessary. We show that they are determined, up to a non-vanishing factor function, by the existence of a system of first-order differential equations providing
two integrals of motion. We generalize the form of that Poisson structures and extend their domain of definition. We apply the theory to the rolling disk,
the Routh's sphere, the ball rolling on a surface of revolution, and its special case of a ball rolling inside a cylinder.

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Author(s): Arturo Ramos
Title: New links and reductions between the Brockett nonholonomic integrator and related systems
Preprint: Rend. Sem. Mat. Univ. Pol. Torino, at press.
YEAR: 2004
(MASIE)Subsection : 1.5
Abstract: We briefly review recently developed techniques for dealing with a class of time-dependent systems defined on Lie groups and homogeneous spaces. The techniques to treat these systems on Lie groups are a generalization of the Wei--Norman method and a reduction theorem. We apply the theory
to the well-known Brockett nonholonomic integrator and to some other related systems: its generalization to second degree and another system with the same
associated Lie algebra, and the generalization to third degree of the nonholonomic integrator. New relations between these systems are found by means of the geometric methods considered.

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Author(s): Arturo Ramos
Title: A connection approach to Lie systems
In: Publicaciones de la RSME 6 (2004), 235--239.
YEAR: 2004
(MASIE)Subsection : Related Publication
Abstract: We review some properties concerning the geometric theory of Lie systems. They can be understood in terms of the geometry of connections in principal and associated bundles. A brief sketch of this relation is given, allowing us to extend, in a natural way, the definition of Lie system to a certain kind of systems of PDEs.

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Author(s): M. Rodríguez-Olmos and M. E. Sousa-Dias.
Title: A Note on the Existence and Symmetries of Relative Equilibria in Simple Mechanical Systems
In: Proceedings of the XI Fall Workshop on Geometry and Physics, Oviedo 2002. Publicaciones de la Real Sociedad Matemática Española, vol. 5, 2004. pp 241-246.
(MASIE)Subsection :1.1
Abstract:A simple mechanical system is a Hamiltonian system of the form "kinetic plus potential'' energy defined on the cotangent bundle of a Riemannian manifold $(Q,\kappa)$. If the Hamiltonian is invariant under the action of a subgroup $G$ of the isometry group of $(Q,\kappa)$, then relative equilibria, i.e. group orbits invariant under the dynamics, can exist. We give a necessary condition for the existence of such solutions independent of the potential and study the symmetries (stabilizers) of these orbits.

2003

Author(s): Henk Broer, Heinz Hanßmann, Àngel Jorba, Jordi Villanueva, Florian Wagener
Title: Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach
Journal: Nonlinearity 16(5), p. 1751-1791
YEAR: 2003
(MASIE)Subsection : 1.1 and 1.2
Abstract: In the conservative dynamics of certain quasi-periodically forced oscillators, normal-internal resonances are considered in a bifurcational setting. So the unperturbed system has one degree of freedom. By averaging, the correspondence is made with the well-known case of periodic forcing and the way in which the present quasi-periodic case complicates the former. This paper extends work on the continuation of normally elliptic tori, where all normal-internal resonances are excluded: presently the gaps in the Cantor set are filled one by one.

Authors: H. Broer, R. Cushman and F. Fasso`
Title: Geometry of KAM tori for nearly integrable Hamiltonian systems
Preprint:
YEAR: 2003
(MASIE)Subsection: 1.2
Abstract: We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangean tori by glueing together local KAM conjugacies with help of a partition of unity. In this way we find a global Whitney smooth conjugacy between a nearly-integrable system and an integrable one. This leads to preservation of geometry, which allows us to define all the nontrivial geometric invariants like monodromy or Chern classes of an integrable system also for nearly integrable systems.
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Author(s): Henk Broer, Heinz Hanßmann, Àngel Jorba, Jordi Villanueva and Florian Wagener
Title: Quasi-Periodic Response Solutions at Normal-Internal Resonances
Preprint: RWTH Aachen
YEAR: 2003
(MASIE)Subsection : 1.1 and 1.2
Abstract: In the conservative dynamics of certain quasi-periodically forced oscillators, normal-internal resonances are considered in a bifurcational setting. The unforced system is a one degree of freedom oscillator, under forcing the system becomes a skew-product flow with a quasi-periodic motion on an $n$-dimensional torus as driving system. In this work, we investigate the persistence and the bifurcations of quasi-periodic $n$-dimensional tori (so-called `response solutions') in the averaged system, filling normal-internal resonance `gaps' that had been excluded in previous analyses.

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Author(s): Henk Broer, Heinz Hanßmann, Jiangong You
Title: Bifurcations of Normally Parabolic Tori in Hamiltonian Systems
Preprint: Rijksuniversiteit Groningen
YEAR: 2003
(MASIE)Subsection : 1.1 and 1.2
Abstract: We consider perturbations of integrable Hamiltonian systems in the neighbourhood of normally parabolic invariant tori. Under appropriate transversality conditions the tori in the unperturbed system bifurcate according to a (generalized) cuspoid catastrophe. Combining techniques of KAM-theory and singularity theory we show that such bifurcation scenarios survive the perturbation on large Cantor sets. Applications to rigid body dynamics and forced oscillators are pointed out.
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Author(s): Jos\'e F. Cariñena and Arturo Ramos
Title: Lie systems in quantum mechanics and control theory
In the book: Classical and Quantum Integrability. Banach Center Publications 59, Polish Academy of Sciences, Warszawa, 2003. Pp: 143-162.
YEAR: 2003
(MASIE)Subsection : Related publication
Abstract: Some simple examples from quantum physics and control theory are used to illustrate the application of the theory of Lie systems. We will show, in particular, that for certain physical models both of the corresponding classical and quantum problems can be treated in a similar way, may be up to the replacement of the involved Lie group by a central extension of it. The geometric techniques developed for dealing with Lie systems are also used in problems of control theory. Specifically, we will study some examples of control systems on Lie groups and homogeneous spaces.
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Author(s): Jos\'e F. Cariñena, Jesús Clemente-Gallardo and Arturo Ramos
Title: Motion on Lie groups and its applications in control theory
Journal: Rep. Math. Phys. 51 (2003), 159-170.
YEAR: 2003
(MASIE)Subsection : Related publication
Abstract: The usefulness in control theory of the geometric theory of motion on Lie groups and homogeneous spaces will be shown. We quickly review some recent results concerning two methods to deal with these systems, namely, a generalization of the method proposed by Wei and Norman for linear systems, and a reduction procedure. This last method allows us to reduce the equation on a Lie group $G$ to that on a subgroup $H$, provided a particular solution of an associated problem in $G/H$ is known. These methods are shown to be very appropriate to deal with control systems on Lie groups and homogeneous spaces, through the specific examples of the planar rigid body with two oscillators and the front-wheel driven kinematic car.
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Author(s): H.R. Dullin, A. Giacobbe, R. Cushman.
Title: Monodromy in the resonant swing spring.
Preprint:
YEAR: 2003
(MASIE)Subsection : 1.2

Abstract: This paper shows that an integrable approximation of the spring pendulum, when tuned to be in $1:1:2$ resonance, has monodromy. The stepwise precession angle of the swing plane of the resonant spring pendulum is shown to be a rotation number of the integrable approximation. Due to the monodromy, this rotation number is not a globally defined function of the integrals. In fact at lowest order it is given by $\arg(a+ib)$ where $a$ and $b$ are functions of the integrals. The resonant swing spring is therefore a system where monodromy has easily observed physical consequences.
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Author(s): H.R. Dullin, S. Vu Ngoc
Title: Vanishing Twist near Focus-Focus Points
Preprint:
YEAR: 2003
(MASIE)Subsection : 1.2

Abstract: We show that near a focus-focus point in a Liouville integrable Hamiltonian system with two degrees of freedom lines of locally constant rotation number in the image of the energy-momentum map are spirals determined by the eigenvalue of the equilibrium. From this representation of the rotation number we derive that the twist condition for the isoenergetic KAM condition vanishes on a curve in the image of the energy-momentum map that is transversal to the line of constant energy. In contrast to this we also show that the frequency map is non-degenerate for every point in a neighborhood of a focus-focus point.
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Author(s): H.R. Dullin, A.V. Ivanov
Title: Vanishing twist in the Hamiltonian Hopf bifurcation
Preprint:
YEAR: 2003
(MASIE)Subsection : 1.2
Abstract: 1: -1 resonance. At the bifurcation the pure imaginary eigenvalues of the elliptic equilibrium turn into a complex quadruplet of eigenvalues and the equilibrium becomes a linearly unstable focus-focus point. We explicitly calculate the frequency map of the integrable normal form, in particular we obtain the rotation number as a function on the image of the energy-momentum map in the case where the fibres are compact. We prove that the isoenergetic non-degeneracy condition of the KAM theorem is violated on a curve passing through the focus-focus point in the image of the energy-momentum map. This is equivalent to the vanishing of twist in a Poincar\'e map for each energy near that of the focus-focus point. In addition we show that in a family of periodic orbits (the non-linear normal modes) the twist also vanishes. These results imply the existence of all the unusual dynamical phenomena associated to non-twist maps near the Hamiltonian Hopf bifurcation.
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Author(s): H.R. Dullin, A.V. Ivanov
Title: (Vanishing) twist in the saddle-centre and period-doubling bifurcation
Preprint:
YEAR: 2003
(MASIE)Subsection : 1.2
Abstract: The lowest order resonant bifurcations of a periodic orbit of a Hamiltonian system with two degrees of freedom have frequency ratio 1:1 (saddle-centre) and 1:2 (period-doubling). The twist, which is the derivative of the rotation number with respect to the action, is studied near these bifurcations. When the twist vanishes the nondegeneracy condition of the (isoenergetic) KAM theorem is not satisfied, with interesting consequences for the dynamics. We show that near the saddle-centre bifurcation the twist always vanishes. At this bifurcation a ``twistless'' torus is created, when the resonance is passed. The twistless torus replaces the colliding periodic orbits in phase space. We explicitly derive the position of the twistless torus depending on the resonance parameter, and show that the shape of this curve is universal. For the period doubling bifurcation the situation is different. Here we show that the twist does not vanish in a neighborhood of the bifurcation.
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Author(s): H. R. Dullin and J. D. Meiss
Title: Twist Singularities for Symplectic Maps
Journal: Chaos, nº1, vol 13, 1-16
YEAR: 2003
(MASIE)Subsection : 1.2
Abstract: Near a nonresonant, elliptic fixed point, a symplectic map can be transformed into Birkhoff normal form. In these coordinates, the
dynamics is represented entirely by the Lagrangian ``frequency map'' that gives the rotation number as a function of the action. The twist matrix, given by the Jacobian of the rotation number, describes the anharmonicity in the system. When the twist is singular the frequency map need not be locally one-to-one. Here we investigate the occurrence of fold and cusp singularities in the frequency map. We show that folds necessarily occur near third order resonances. We illustrate the results by numerical computations of frequency maps for a quadratic, symplectic map.

Author(s): H. R. Dullin and G. Gottwald and D.D. Holm
Title: Camassa-Holm, Korteweg-de~Vries-5 and other asymptotically equivalent equations for shallow water waves
Journal: Fluid Dynamics Research (to appear)
YEAR: 2003
(MASIE)Subsection : 4.1
Abstract: The integrable 3rd-order Korteweg-de~Vries (KdV) equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves. However, at quadratic order, this asymptotic expansion produces an entire {\it family} of shallow water wave equations that are asymptotically equivalent to each other, under a group of nonlinear, nonlocal, normal-form transformations introduced by Kodama. These Kodama transformations are used to present connections between shallow water waves, the integrable 5th-order Korteweg-de~Vries equation, and a generalization of the Camassa-Holm (CH) equation that contains an additional integrable case. The CH travelling wave solutions are classified as a function of Bond number by using phase plane analysis. Finally, we discuss the experimental observability of the CH solutions.

Author: F. Fasso`
Title: Comparison of splitting algorithms for the rigid body
Journal: Journal of Computational Physics 189, 527-538 (2003)
YEAR: 2003
(MASIE)Subsection: 2.2
Abstract: We compare several different second order splitting algorithms for the asymmetric rigid body, with the aim of determining which one produces the smallest energy error for a given rigid body, namely, for given moments of inertia. The investigation is based on the analysis of the dominant term of the modified Hamiltonian and indicates that different algorithms can produce energy errors which differ by several orders of magnitude. As a byproduct of this analysis we remark that, for the special case of a flat rigid body with moments of inertia proportional to $(1,0.75,0.25)$, one of the considered algorithms is in fact of order four.
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Author: David J. Fernández C., Rodrigo Muñoz and Arturo Ramos
Title: Second order SUSY transformations with `complex energies'
Journal: Phys. Lett. A 308 (2003), 11-16.
YEAR: 2003
(MASIE)Subsection: Related publication
Abstract: Second order supersymmetry transformations which involve a pair of complex conjugate factorization energies and lead to real non-singular potentials are analyzed. The generation of complex potentials with real spectra is also studied. The theory is applied to the free particle, one-soliton well and one-dimensional harmonic oscillator.
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Author(s): Heinz Hanßmann and Jan-Cees van der Meer
Title: On Non-Degenerate Hamiltonian Hopf Bifurcations in 3DOF Systems
Preprint: RWTH Aachen
YEAR: 2003
(MASIE)Subsection : 1.1 and 1.2
Abstract: We give a short review of available methods to determine the non-degeneracy of Hamiltonian Hopf bifurcations in three-degree-of-freedom systems. We illustrate the geometric
method to more detail, using the example of the Lagrange top.

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Author(s): Heinz Hanßmann
Title: Hamiltonian Bifurcations of Invariant Tori with a Vanishing Floquet Exponent
Preprint: RWTH Aachen
YEAR: 2003
(MASIE)Subsection : 1.1 and 1.2
Abstract: Local bifurcations of invariant tori are induced by the normal behaviour and occur where the latter changes from elliptic to hyperbolic. For invariant tori in Floquet form this is described by the Floquet exponents. When one of the Floquet exponents vanishes even the persistence of the bifurcating tori themselves is in question. With the actions conjugate to the toral angles serving as unfolding parameters, one can look for persistence of the pertinent bifurcation scenario instead. Such a study lies at the intersection of KAM theory and singularity theory. This paper presents the results of current research and ends with a speculation on how far the borders may be pushed.
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Author(s): Heinz Hanßmann
Title: Hamiltonian Torus Bifurcations Related to Simple Singularities
Preprint: RWTH Aachen
YEAR: 2003
(MASIE)Subsection : 1.1 and 1.2
Abstract: Lower dimensional $n$-tori in integrable and nearly integrable Hamiltonian systems with $n+1$ degrees of freedom are considered. The one-degree-of-freedom dynamics normal to the invariant tori is governed by (the singularities of) a family of planar Hamiltonian functions. The ensuing bifurcation scenarios are shown to survive both integrable and non-integrable perturbations.
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Author(s): Heinz Hanßmann, Jan-Cees van der Meer
Title: Algebraic methods for determining Hamiltonian Hopf bifurcations in three-degree-of-freedom systems
Preprint: Technische Universiteit Eindhoven
YEAR: 2003
(MASIE)Subsection : 1.1 and 1.2
Abstract: When considering bifurcations, the type of bifurcation is usually classified by comparing to standard situations or normal forms. It is shown how Hamiltonian Hopf bifurcations can be determined in three-degree-of-freedom systems, as is done in this paper for the $3D$~H\'enon-Heiles family. After a careful formulation of the local once reduced system in terms of properly chosen invariants the system can be compared to the standard form to determine the presence of non-degenerate Hamiltonian Hopf bifurcations.
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Authors: James Montaldi and Tadashi Tokieda
Title: Openness of momentum maps and persistence of extremal    relative equilibria
Journal: Topology 42, 833-844
Year: 2003
(MASIE)Subsection: 1.1
Abstract:   We prove that for every proper Hamiltonian action of a Lie group G in finite dimensions the momentum map is locally G-open relative to its image (i.e.\ images of G-invariant open sets are open). As an application we deduce that in a Hamiltonian system with continuous Hamiltonian symmetries, extremal relative equilibria persist for every perturbation of the value of the momentum map, provided the isotropy subgroup of this value is compact. We also demonstrate how this persistence result applies to an example of ellipsoidal figures of rotating fluid, and provide an example with plane point vortices which shows how the compactness assumption is related to persistence.
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Authors: J Montaldi, A Souliere and T. Tokieda
Title: Point vortices on a cylinder
Journal: SIAM J. on Applied Dynamical Systems (to appear - 2003)
Year: 2003
(MASIE)Subsection: 1.1 and 4.2
Abstract:   Point vortices on a cylinder (periodic strip) are studied geometrically. The Hamiltonian formalism is developed, a non-existence theorem for relative equilibria is proved, equilibria are classified when all vorticities have the same sign, and several results on relative periodic orbits are established, including as corollaries classical results on vortex streets and leapfrogging.
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Authors: Matthew Perlmutter, Miguel Rodriguez-Olmos, M. Esmeralda Sousa-Dias
Title: On the geometry of reduced cotangent bundles at zero momentum
Preprint: math.SG/0310437
Year: 2003
(MASIE)Subsection: 1.1
Abstract: We consider the problem of cotangent bundle reduction for non free group actions at zero momentum. We show that in this context the symplectic stratification obtained by Sjamaar and Lerman refines in two ways: (i) each symplectic stratum admits a stratification which we call the secondary stratification with two distinct types of pieces, one of which is open and dense and symplectomorphic to a cotangent bundle; (ii) the reduced space at zero momentum admits a finer stratification than the symplectic one into pieces that are coisotropic in their respective symplectic strata.
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Author(s): H. Waalkens and A. Junge and H. R. Dullin
Title: Quantum Monodromy in the Two-Centre Problem
Journal: J. Phys. A, (to appear)
YEAR: 2003
(MASIE)Subsection : 3.3
Abstract: Using modern tools from the geometric theory of Hamiltonian systems it is shown that electronic excitations in diatoms which can be modelled by the two centre problem exhibit a complicated case of classical and quantum monodromy. This means that there is an obstruction to the existence of global quantum numbers in these classically integrable systems. The symmetric case of $H_2^+$ and the asymmetric case of $HHe^{++}$ are explicitly worked out. The asymmetric case has a non-local singularity causing monodromy. It coexists with a second singularity which is also present in the symmetric case. An interpretation of monodromy is given in terms of the caustics of invariant tori.

2002

Authors: L. Allen & T.J. Bridges
Title:   Numerical exterior algebra and the compound matrix method
Journal: Numerische Mathematik 92: 197-232
Year:    2002
(MASIE) Subsection: 2.2
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Authors: P. Ashwin, M.V. Bartuccelli, T.J. Bridges & S.A. Gourley
Title:   Travelling fronts for the KPP equation with spatio-temporal delay
Journal: Z. angew. Mathematik & Physik 53: 103-122
YEAR:    2002
(MASIE) Subsection: 4.5
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Author(s): E. Barth, B. Leimkuhler and S. Reich
Title: A test set for molecular dynamics.
Journal: to appear in Springer Lecture Notes in CSE.
YEAR: 2002
(MASIE)Subsection : 2.1

Author(s): M.V. Bartuccelli, G. Gentile and K. Georgiou
Title: On the stability of the upside-down pendulum with damping.
Journal: Proceedings of the Royal Society of London, Series A, 458, 255-269.
YEAR: 2002
(MASIE)Subsection : 1.3
Abstract: A rigorous analysis is presented in order to show that, in presence of friction, the upward equilibrium position of the vertically driven pendulum, with a small non-vanishing damping term, becomes asymptotically stable when the period of the forcing is below an appropriate threshold value. As a byproduct we obtain an analytic expression of the solution for initial data close enough to the equilibrium position.

Author(s): M.V. Bartuccelli, G. Gentile
Title: Lindstedt series for perturbations of isochronous systems: a review of the general theory.
Journal: Reviews in Mathematical Physics, 14, 121 - 171.
YEAR: 2002
(MASIE)Subsection : 1.2
Abstract: We give a review of the general theory of perturbations of isochronous systems by giving a proof of the persistence of invariant
tori for analytic perturbations of isochronous systems by using the Lindstedt series expansion for the solutions of the equations of motion.

Author(s): G. Benettin, A.M. Cherubini and F. Fassò
Title: Regular and chaotic motions of the fast rotating rigid body: a numerical study
Journal: Discrete and Continuous Dynamical Systems, vol. 2, 521-540
YEAR: 2002
(MASIE)Subsection: 1.2
Abstract: We numerically investigate the dynamics of a symmetric rigid body with a fixed point in a small analytic external potential
(equivalently, a fast rotating body in a given external field) in the light of previous theoretical investigations based on Nekhoroshev
theory. Special attention is posed on ``resonant'' motions, for which the tip of the unit vector in the direction of the angular momentum vector can wander, for no matter how small the perturbation is, on an extended, essentially two--dimensional, region of the unit sphere, a phenomenon called ``slow chaos''. We produce numerical evidence that slow chaos actually takes place in simple cases, in agreement with the theoretical prediction. Chaos however disappears for motions near proper rotations around the symmetry axis, thus indicating that the theory of these phenomena still needs to be improved. A heuristic explanation is proposed.

Author(s): Birtea, P., M. Puta, and T.S. Ratiu
Title: Controllability of reduced systems
Preprint:
YEAR: 2002
(MASIE)Subsection: Related Publication

Author(s): Birtea, P., M. Puta, T.S. Ratiu, and R. Tudoran
Title: A short proof of chaos in an atmospheric system
Preprint:
YEAR: 2002
(MASIE)Subsection: 1.4

Author(s): Bloch, A.M., P.E. Crouch, J.E. Marsden, and T.S. Ratiu
Title: The symmetric and discrete rigid body equations
Journal: Nonlinearity (to appear)
YEAR: 2002
(MASIE)Subsection: 2.2

Author(s): K.B. Blyuss
Title: Chaotic behaviour of solutions to a perturbed Korteweg-de Vries equation
Journal: Reports on Mathematical Physics
YEAR: 2002
(MASIE)Section: 1
Abstract: Stationary wave solutions of the perturbed Korteweg-de Vries equation are considered in the presence of external hamiltonian
perturbations. Conditions of their chaotic behaviour are studied with the help of Melnikov theory. For the homoclinic chaos Poincaré
sections are constructed to demonstrate the complicated behaviour, and Lyapunov exponents are also numerically calculated.

Authors: K.B. Blyuss, T.J. Bridges & G. Derks
Title:   Transverse instability and its long-term development for solitary waves of the (2+1)-Boussinesq equation.
Journal: Preprint
Year:    2002
(MASIE) Subsection: 4.5

Authors: T.J. Bridges & K.V. Georgiou
Title:   Computing global orbits of the forced spherical pendulum
Journal: PhysicaD 165: 1-11
Year:    2002
(MASIE) Subsection: 2.2
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Author(s): Thomas J. Bridges, Gianne Derks and Georg Gottwald
Title: Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework
Journal: To appear in Physica D.
YEAR: 2002
(MASIE)Subsection: 2.3 and 4.5
Abstract: The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation
is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. A numerical framework, based on a robust shooting algorithm on exterior algebra spaces is introduced. The complete algorithm has several novel features, including a rigorous algorithm for choosing starting values, the role of Grassmannian submanifolds in choosing the
numerical integrator, and the use of the Hodge star operator for deducing a numerically computable form for the Evans function.   The algorithm is illustrated by computing the stability and instability of solitary waves of the fifth-order KdV equation with polynomial nonlinearity.
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Author(s): T.J. Bridges and G. Derks
Title: Linear instability of solitary wave solutions of the Kawahara equation and its generalizations
Journal: SIAM J. Math. Anal. 33, pp 1356-1378
YEAR: 2002
(MASIE)Subsection : 4.5
Abstract: The linear stability problem for solitary-wave states of the Kawahara - or fifth-order KdV-type - equation and its generalizations is considered. A new formulation of the stability problem in terms of the symplectic Evans matrix is presented. The formulation is based on a multi-symplectification of the Kawahara equation, and leads to a new characterization of the basic solitary wave, including changes in the state
at infinity represented by embedding the solitary wave in a multi-parameter family. The theory is used to give a rigorous geometric sufficient condition for instability. The theory is abstract and applies to a wide range of solitary-wave states. For example, the theory is applied to the families of solitary waves found by Kichenassamy-Olver and Levandosky.
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Author(s): T.J. Bridges and G. Derks
Title: Constructing the symplectic Evans Matrix using maximally analytic individual vectors
Journal: To appear in Roy. Soc. Edinburgh Proc. A .
YEAR: 2002
(MASIE)Subsection : 4.5
Abstract: For linear systems with a multi-symplectic structure, arising from the linearization of Hamiltonian PDEs about a solitary wave, the Evans function can be characterized as the determinant of a matrix, and each entry of this matrix is a restricted symplectic form. However, in general this matrix of two-forms may have branch points at isolated points, shrinking the natural region of analyticity.
In this paper, a new construction of the symplectic Evans matrix is presented which is based on individual vectors but is
analytic at the branch points -- indeed maximally analytic. In fact this result has greater generality than just the symplectic case: it solves
the following open problem in the literature: can the Evans function be constructed in a maximally analytic way when
individual vectors are used?    Although the non-symplectic case will be discussed in passing, the paper will concentrate on the symplectic case, where there are geometric reasons for evaluating the Evans function on individual vectors.
This result simplifies and generalizes the multi-symplectic framework for the stability analysis of solitary waves, and some of the implications are discussed.
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Authors: T.J. Bridges
Title:   Stability of solitary waves: geometry, symplecticity and three-dimensionality
Journal: Benjamin Memorial Lecture, in Proceedings of 2001 IMA WOW Conference
Year:    2002
(MASIE) Subsection: 4.5

Authors: T.J. Bridges & G. Derks
Title:   Dimension breaking of gradient elliptic operators
Journal: Conference on Nonlinear Analysis 2002 (in honour of K. Kirchgaessner's 70th birthday)
Year:    2002
(MASIE) Subsection: 4.5

Author(s): Busuioc, V. and T.S. Ratiu
Title: The second grade fluid and averaged Euler equations with Navier-slip boundary conditions
Preprint:
YEAR: 2002
(MASIE)Subsection: 2.1

Author(s): J. F. Cariñena and A. Ramos
Title: A new geometric approach to Lie systems and physical applications
Journal: Acta Applicandae Mathematicae 70, 43-69.
YEAR: 2002
(MASIE)Subsection: Related publication
Abstract:   The characterization of systems of differential equations admitting a superposition function allowing us to write the general
solution in terms of any fundamental set of particular solutions is discussed. These systems are shown to be related with equations on
a Lie group and with some connections in fiber bundles. We develop two methods for dealing with such systems: the generalized Wei-Norman method and the reduction method, which are very useful when particular solutions of the original problem are known. The theory is illustrated with some applications in both classical and quantum mechanics.
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Author(s): J. F. Cariñena and A.Ramos
Title: Lie systems in Control Theory
Journal: in Nonlinear Geometric Control Theory, World Scientific, Singapore, 287-304.
YEAR: 2002
(MASIE)Subsection: Related publication
Abstract: We show that the theory of systems of differential equations allowing a rule for expressing the general solution in terms
of some particular ones, the so-called Lie systems, are relevant in Control Theory. These systems are related to some particular cases
of control systems on Lie groups, whose exact solution is usually found by means of a method generalizing the one proposed by Wei
and Norman. We establish some results concerning these systems. The general theory is illustrated with some examples.
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Author(s): J. F. Cariñena and Arturo Ramos
Title: Lie-Scheffers systems in Physics
Journal: in Recent Advances in Lie Theory, (Research and Exposition in Mathematics Vol. 25), Heldermann Verlag, Berlin, 181-188.
YEAR: 2002
(MASIE)Subsection: Related publication
Abstract:   We recall the Theorem by Lie and Scheffers concerning the characterization of systems of differential equations admitting
a superposition function, i.e., those whose general solution can be written in terms of some particular solutions and constants.
Each of these systems is related with a Lie algebra, specified by the own Theorem. We expose some recently developed Lie theoretic and
geometric techniques, useful for treating such systems, as a reduction property and a generalization of the Wei-Norman method. We illustrate the theory with some applications, which are mainly inspired in physical problems.
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Author(s): Castrillón López, M., J. J. Muñoz Masqué, and T.S. Ratiu
Title: Gauge invariance and variational trivial problems on the bundle of connections
Journal: Diff. Geom. Appl. (to appear)
YEAR: 2002
(MASIE)Section: 4

Author(s): Castrillón López and T.S. Ratiu
Title: Reduction in principal bundles: Covariant Lagrange-Poincaré equations
Preprint:
YEAR: 2002
(MASIE)Section: 4

Author(s): Cendra, H., J.E. Marsden, and T.S. Ratiu
Title: Cocycles and compatibility
Preprint:
YEAR: 2002
(MASIE)Section: 1

Editors: A. Chenciner, R. Cushman, C. Robinson, Z. Xia
Title:``Celestial Mechanics''
Serial book: Contemporary Mathematics, vol. 292
Year: 2002
Publisher: American Mathematical Society, Providence, R.I. - ISBN 08218-2902-5
(MASIE)Subsection:1.2
Notes: Proceedings of a conference on celestial mechanics, Evanston, Ill. 1999.

Author(s): Chossat, P., J.-P. Ortega, and T.S. Ratiu
Title: Hamiltonian Hopf bifurcation with symmetry
Journal: Arch. Rat. Mech. Anal., 163, 1--33.
YEAR: 2002
(MASIE)Subsection: 1.1

Author(s): Chossat, P., D.K. Lewis, J.-P. Ortega, and T.S. Ratiu
Title: Bifurcation of relative equilibria in mechanical systems with symmetry
Journal: Advances in Applied Math. (to appear)
YEAR: 2002
(MASIE)Subsection: 1.1

Author(s): Chossat, P.
Title: The reduction of equivariant dynamics to the orbit space for compact group actions.
Journal: Acta Applicandae Mathematicae, 70, 71--94.
YEAR: 2002
(MASIE)Section: 1

Author(s): Chossat, P. and Armbruster, D
Title: Dynamics of polar reversals in spherical dynamics.
Journal: To appear in Proc. R. Soc. London.
YEAR: 2002
(MASIE)Section: 1

Author(s): G. Cicogna and S. Walcher
Title: Convergence of normal form transformation: The role of symmetries
Journal: Acta Appl. MAth. 70, 95-111
YEAR: 2002
(MASIE)Section: 1

Author(s): Clarke, S., Malomed, B.A. and Grimshaw, R.
Title: ``Dispersion management" for solitons in a Korteweg-de Vries system.
Journal: Chaos, 12, 8-15.
YEAR: 2002
(MASIE)Subsection : 4.1

Author(s): R. Cushman and L. Bates
Title: The odd symplectic group in geometry
Preprint: University of Calgary
Year: June 2002
(MASIE)Section: 1
Abstract: We describe the relation of the odd symplectic group to contact geometry and its use in the study of periodic geodesics in Riemannian geometry.

Author(s): R. Cushman and B. Westbury
Title: Conjugacy classes in the odd symplectic group
Preprint: University of Warwick
Year: May 2002
(MASIE)Section: 1
Abstract: In this paper we give a representative of an orbit of an element of the odd symplectic group under conjugation. We follow the ideas of Burgoyne and Cushman (J. Alg. vol. 44 (1977), 333-362) and concentrate on adjoint orbits in the Lie algebra. There are orbits which have moduli that are not eigenvalues.

Author(s): R. Cushman and B.I. Zhilinskii
Title: Monodromy of a two degree of freedom Liouville integrable system with many focus-focus singular points
Journal: (to appear in J. Phys. A)
Year: 2002
(MASIE)Subsection: 1.2
Abstract: This paper deals with the global monodromy of singular Lagrangian toral fibrations defined by two degree of freedom Liouville integrable systems with only focus-focus singular points. We show that any global monodromy matrix in the set of two by two matrices with integer entries of determinant one is realizable by such a system.

Author(s):R. Cushman and J. Sniatycki
Title: Nonholonomic reduction for free and proper actions
Journal: Regular and Chaotic Dynamics, vol. 7, 61--72
Year: 2002
(MASIE)Subsection: 1.5
Abstract: We study a nonholonomically constrained Hamiltonian system with a symmetry group which acts properly and freely on a constraint distribution. We show that the reduced dynamics is described by a generalized distributional Hamiltonian system. The general theory is illustrated by the example of Chaplygin's skate.

Author(s): R. Cushman and San Vu Ngoc
Title: Sign of the monodromy for Liouville integrable systems
Journal: to appear in Ann. Inst. H. Poincaré
Year: 2002
(MASIE)Subsection: 1.2
Abstract: In this note we show that the monodromy of a two degree of freedom integrable Hamiltonian system has a universal sign in
the case of a focus-focus singularity. We also show how to extend the monodromy index to several focus-focus points when the integrable system has a circular symmetry.

Author(s):R. Cushman and J. \'{S}niatycki
Title: A nonholonomic oscillator
Journal: (to appear in Rep. Math. Phys.)
Year: 2002
(MASIE)Subsection:1.5
Abstract: In this paper we carry out singular reduction for the noholonomic oscillator and compare it with results obtained by applying the momentum equation. The momentum equation is does not give the full equations of motion.

Author(s):R. Cushman and M.R. Roberts
Title: Poisson structures transverse to coadjoint orbits
Journal: (to appear in Bull. Sci. Math.)
Year: 2002
(MASIE)Section: 1
Abstract:We show that the Poisson structure transverse to a coadjoint orbit in the dual of a semisimple Li
algebra has a polynomial structure matrix, as conjectured by Damianou (Bull. Sci. Math., 120, 1996, p. 195-214).

Author(s): R. Cushman and F. Beukers
Title: The complex geometry of the spherical pendulum
Journal: Contemporary Mathematics, vol. 292
Year: 2002
(MASIE)Subsection: 1.2
Abstract: In this paper we describe the geometry of the energy momentum mapping of the complexified spherical pendulum. For background on the classical spherical pendulum we refer the reader to chapter 4 of Global Aspects of Classical Integrable Systems by Cushman and Bates. We show that this complex Hamiltonian system is Mumford-Jacobi completely integrable and admits a complex analogue of the period lattice and action angle coordinates. Following the variation of the period lattice around a closed loop in the complement of the discriminant locus of a family of elliptic curves, we show that the complexified spherical pendulum has monodromy.

Author(s): G.Derks and T. Ratiu
Title: Unstable manifolds of relative equilibria in Hamiltonian systems with dissipation
Journl: Nonlinearity, 15, 531--549 .
YEAR: 2002
(MASIE)Subsection: 1.1 and 1.3
Abstract: This paper studies the destabilizing effects of dissipation on families of relative equilibria in Hamiltonian systems which are non-extremal constraint critical points in the energy-Casimir or the energy-momentum methods. The dissipation is allowed to destroy the
conservation law associated with the symmetry group or Casimirs, as long as the family of relative equilibria stays on an invariant manifold. This approach complements earlier work in the literature, in which the dissipation did not affect the conservation law.
First, Chetaev's instability theorem is extended to invariant manifolds and this extended theorem is used to prove instability of families of relative equilibria for several examples. Second, it is shown that families of non-extremal stationary solutions of the two-dimensional incompressible homogeneous Euler equations are unstable for the corresponding viscous perturbations of this system, that is, for the two-dimensional Navier-Stokes equations. Also, the instability of the sleeping top relative equilibria under friction can easily be proved in this way, even before the Hamiltonian sleeping top becomes linearly unstable. Finally, sufficient conditions are given for which friction destabilizes families of non-extremal relative equilibria in simple mechanical systems with abelian symmetry.
URL

Author(s): Dr\u agulete, O., L. Ornea, and T.S. Ratiu
Title: Reduction of cosphere bundles
Preprint:
YEAR: 2002
(MASIE)Section: 1

Author(s): H. Dullin and F. Fasso`
Title: An algorithm for detecting Directional Quasi-Convexity
Preprint:
YEAR: 2002
(MASIE)Subsection: 1.1
Abstract: Directional Quasi--Convexity (DQC) is a sufficient condition for Nekhoroshev stability, that is, stability for finite but very long times, of elliptic equilibria of Hamiltonian systems.
The numerical detection of DQC is elementary for system with three degrees of freedom. In this article, we propose a recursive algorithm to test DQC in any number $n\ge4$ of degrees of freedom.
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Author(s): H. R. Dullin, J. E. Howard, and M. Hor`anyi.
Title: Generalizations of the Störmer problem for dust grain orbits.
Journal: Physica D, 171:178–195
YEAR: 2002
(MASIE)Subsection : 1.1
Abstract: We investigate the generalized St¨ormer Problem, which includes electromagnetic and gravitational forces on a charged dust grain near an axisymmetric planet. For typical charge to mass ratios neither force can be neglected. The effects of the different forces are discussed in detail. Thus, including the gravitational force gives rise to stable circular orbits lying in a plane entirely above/below the equatorial plane. A modified 3rd Kepler's Law for these orbits is found and analyzed.
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Author(s): El, G.A. and Grimshaw, R.
Title: Generation of undular bores in the shelves of slowly-varying solitary waves.
Journal: Chaos (to appear)
YEAR: 2002
(MASIE)Subsection : 4.1

Author(s):   F. Fassò and A. Giacobbe
Title: Geometric structure of "broadly integrable" Hamiltonian systems
Journal: Journal of Geometry and Physics, Vol. 44, 156-170
YEAR: 2002
(MASIE)Subsection: 1.2
Abstract: We study the geometry of the fibration in invariant tori of a Hamiltonian system which is integrable in Bogoyavlenkij's ``broad sense''---a generalization of the standard cases of Liouville and noncommutative integrability. We show that the structure of such a fibration generalizes that of the standard cases. Firstly, the base manifold has a Poisson structure. Secondly, there is a natural way of arranging the invariant tori which generates a second foliation of the phase
space; however, such a foliation is not just the polar to the invariant tori. Finally, under suitable conditions, there is a notion of an "action manifold'' with an affine structure. We also study the analogous of the problem of the existence of "global action--angle coordinates'' for these systems.
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Author(s): S. Ferrer, H. Hanßmann, J. Palacián and P. Yanguas
Title: On Perturbed Oscillators in 1-1-1 Resonance : The Case of Axially Symmetric Cubic Potentials
Journal: J. Geom. Phys. 40, p. 320 -- 369
YEAR: 2002
(MASIE)Subsection : 1.2
Abstract: Axially symmetric perturbations of the isotropic harmonic oscillator in three dimensions are studied. A normal form transformation introduces a second symmetry, after truncation. The reduction of the two symmetries leads to a one-degree-of-freedom system. We use a special set of action-angle variables, as well as conveniently chosen generators of the ring of invariant functions. Both approaches are compared and their advantages and disadvantages are pointed out. The reduced flow of the normal form yields information on the original system. We illustrate the results by analysing the family of (arbitrary) axially symmetric cubic potentials.

Author(s): G. Gaeta, and S. Walcher
Title: Lie Algebras with finite dimensional polynomial centralizer
Journal: J. MAth. Anal. Appl. 269, 578-587.
YEAR: 2002
(MASIE)Section: 1

Author(s): G. Gottwald, J. Frank and S.Reich
Title: A Hamiltonian particle-mesh method for the rotating shallow water equations.
Journal: Meshless Methods, Lecture Notes in Computational Science an Engineering
YEAR: Springer (2002)
(MASIE)Subsection : 2.3
Abstract: A new particle-mesh method is proposed for the rotating shallow-water equations. The spatially truncated equations are Hamiltonian and satisfy a Kelvin circulation theorem. The generation of non-smooth components in the layer-depth is avoided by applying a smoothing operator similar to what has recently been discussed in the context of $\alpha$-Euler models.

Author(s): G. Gottwald, R. Grimshaw and B. Malomed
Title: Cuspons, peakons and regular gap solitons between three dispersion curves.
Journal: Phys. Rev. E (to appear)
YEAR: 2002
(MASIE)Subsection : 4.5
Abstract: A general model is introduced to describe a wave-envelope system for the situation when the linear dispersion relation has three branches, which in the absence of any coupling terms between these branches, would intersect pair-wise in three nearly-coincident points. The system contains two waves with a strong linear coupling between them, to which a third wave is
then coupled. This model has two gaps in its linear spectrum. As is typical for wave-envelope systems, the model also contains a set of cubic nonlinear terms. Realizations of this model can be made in terms of temporal or spatial evolution of optical fields in, respectively, either a planar waveguide, or a bulk-layered medium resembling a photonic-crystal fiber, which carry a triple spatial Bragg grating. Another physical system described by the same general model is a set of three internal wave modes in a density-stratified fluid, whose phase speeds come into close coincidence for a certain wavenumber. A nonlinear analysis is performed for zero-velocity solitons, that is, they have zero velocity in the reference frame in which the third wave has zero group velocity. If one may disregard the self-phase modulation (SPM) term in the equation for the third wave, we find an analytical solution which shows that there simultaneously exist two different families of solitons: regular ones, which may be regarded as a smooth deformation of the usual gap solitons in a two-wave system, and cuspons, which have finite amplitude and energy, but a singularity in the
first derivative at their center. Even in the limit when the linear coupling of the third wave to the first two nearly vanishes, the soliton family remains drastically different from that in the uncoupled system; in this limit, regular solitons whose amplitude exceeds a certain critical value are replaced by peakons. While the regular solitons, cuspons, and peakons are found in an exact analytical form, their stability is tested numerically, which shows that they all may be stable. If the SPM terms are retained, we find that there may again simultaneously exist two different families of generic stable soliton solutions, namely, regular ones and peakons. Direct simulations show that both types of solitons are stable in this case.
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Author(s): G. Gottwald, R. Grimshaw and B. Malomed
Title: Cuspons and peakons vis-a-vis regular solitons and collapse in a three-wave system.
Journal: Proceedings of the AMS-IMS-SIAM Conference "The Legacy of Inverse Scattering Theory in Nonlinear Wave Propagation CONM (Contemporary Math) AMS series(to appear)
YEAR: 2002
(MASIE)Subsection : 4.5
Abstract: We introduce a general model of a one-dimensional three-component wave system with cubic nonlinearity. Linear couplings between the components prevent intersections between the corresponding dispersion curves, which opens two gaps in the system's linear spectrum. Detailed analysis is performed for zero-velocity solitons, in the reference frame in which the group velocity of one wave is zero. Disregarding the self-phase-modulation (SPM) term in the equation for that wave, we find an analytical solution which shows that there simultaneously exist two different families of generic solitons: regular ones, which may be regarded as a smooth deformation of the usual gap solitons in the two-wave system, and cuspons with a singularity in the first derivative at the center, while their energy is finite. Even in the limit when the linear coupling of the zero-group-velocity wave to the other two components is vanishing, the soliton family remains drastically different from that in the linearly uncoupled system: in this limit, regular solitons whose amplitude exceeds a certain critical value are replaced by peakons. While the regular solitons, cuspons, and peakons are found in an exact analytical form, their stability is tested numerically, showing that they all may be stable. In the case when the cuspons are unstable, the instability may trigger onset of spatio-temporal collapse in the system. If the SPM terms are
retained, we find that there again simultaneously exist two different families of generic stable soliton solutions, which are regular ones and peakons. The existence of the peakons depends, in this case, on the sign of certain parameters of the system. Direct simulations show that both types of the solitons may be stable in this most general case too.
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Author(s): Grimshaw, R., Malomed, B. and Gottwald, G.
Title: Singular and regular gap solitons between three dispersion curves.
Journal: Phys. Rev. E (to appear)
YEAR: 2002
(MASIE)Subsection : 4

Author(s): Grimshaw, R., Pelinovsky, D., Pelinovsky, E. and Slunyaev, A.
Title: The generation of large-amplitude solitons from an initial disturbance in the extended Korteweg-de Vries equation.
Preprint:
YEAR: 2002
(MASIE)Subsection : 4

Author(s): Grimshaw, R. and Pelinovsky, E.
Title: Interaction of a solitary wave with an external force in the extended Korteweg-de Vries equation.
Journal: Bifurcation and Chaos (to appear)
YEAR: 2002
(MASIE)Subsection : 4.1

Author(s): Grimshaw, R. and Iooss, G.
Title: Solitary waves of a coupled Korteweg-de Vries system
Journal: Mathematics and Computers in Simulation (to appear)
YEAR: 2002
(MASIE)Subsection : 4.1

Author(s): Grimshaw, R., Pelinovsky, E. and Poloukhina, O.
Title: Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface.
Journal: Nonlinear Processes in Geophysics (to appear)
YEAR: 2002
(MASIE)Subsection : 4.1

Author(s): Grimshaw, R. and Skyrnnikov, Y.
Title: Long-wave instability in a three-layer stratified shear flow.
Journal: Stud. Appl. Math., 108, 77-88.
YEAR: 2002
(MASIE)Subsection : 4.1

Author(s): H. Hanßmann and Jan-Cees van der Meer
Title: On the Hamiltonian Hopf bifurcations in the 3D Henon-Heiles family
Journal: J. Dynamics Diff. Eq. 14, p. 675 -- 695
YEAR: 2002
(MASIE)Subsection : 1.1 and 1.2
Abstract: An axially symmetric perturbed isotropic harmonic oscillator undergoes several bifurcations as the parameter $\lambda$ adjusting the relative strength of the two terms in the cubic potential is varied. We show that three of these bifurcations are Hamiltonian Hopf bifurcations. To this end we analyse an appropriately chosen normal form. It turns out that the linear behaviour is not that of a typical Hamiltonian Hopf bifurcation as the eigenvalues completely vanish at the bifurcation. However, the nonlinear structure is that of a Hamiltonian Hopf bifurcation. The result is obtained by formulating geometric criteria involving the normalized Hamiltonian and the reduced phase space.

Author(s): J.K. Hale, L.T. Magalhães and W.M. Oliva
Title: Dynamics in Infinite Dimensions
Book: 2nd edition, AMS series n.47, Springer- Verlag
YEAR: 2002
(MASIE)Section : Related publication

Author(s): Holm, D.D., J.E. Marsden, and T.S. Ratiu
Title: Euler--Poincaré equations in geophysical fluid dynamics
Journal: Proceedings of the Isaac Newton Institute(to appear)
YEAR: 2002
(MASIE)Section : 4

Author(s): I. Hoveijn, J.S.W. Lamb and R.M. Roberts
Title: Normal forms and unfoldings in eigenspaces of (anti-) automorphisms
Journal: J. Differential Equations (to appear)
YEAR: 2002
(MASIE)Subsection : 1.1
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Author(s): Ionescu D.
Title: Can BE Conserved the Concept of Homogeneozs Gravitational Field from Classical Mechanics in the Relativistic Theory of
Gravitation?
Journal: Theoretical and Mathematical Physics vol. 130 (2), 287-297.
YEAR: 2002
(MASIE)Section: 1

Author(s): Ionescu D.
Title: Comparative Analysis of the Electrogravitational Kepler Problem in GRT and RTG
Journal: Intern. J. of Non-linear Mechanics ( to appear).
YEAR: 2002
(MASIE)Section: 1

Author(s): Ionescu D.
Title: The Electrogravitational Field of an Electrically Charged Mass Point and the Causality Principle in RTG
Preprint
YEAR: 2002
(MASIE)Section: 1

Author(s): Ionescu D. and Soós E.
Title: Simultateity and non-holonomy
Preprint
YEAR: 2002
(MASIE)Section: 1

Author(s): M. Kobayashi and W.M. Oliva
Title: On the Birkhoff approach to Classical Mechanics
Preprint: CAMGSD and D.Mecanica IST- Lisbon
YEAR: 2002
(MASIE)Section : 1

Author(s): H.-P. Kruse, J. Scheurle, T.S. Ratiu
Title: Radial solutions of a two-dimensional version of the Camassa-Holm equation
Preprint
YEAR: 2002
(MASIE)Section: 4

Author(s): J. Lamb, C. Wulff
Title: Reversible relative periodic orbits
Journal: J. Diff. Eq. , 178, 60-100
YEAR: 2002
(MASIE)Subsection : 1.1
Abstract: We study the bundle structure near reversible relative periodic orbits in reversible equivariant systems. In particular we show that the vector field on the bundle forms a skew product system, by which the study of bifurcation from reversible relative periodic solutions reduces to the analysis of bifurcation from reversible discrete rotating waves. We also discuss possibilities for drifts along group orbits.
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Author(s): R. Lasser, and S. Walcher
Title: Error estimates for linear compartmental systems
Journal: SIAM J. Matrix Analysis Appl. 23 (4), 1013-1024.
YEAR: 2002
(MASIE)Section: 1

Author(s): F. Laurent-Polz
Title: Point vortices on the sphere: a case with opposite vorticities
Journal: Nonlinearity, 15, 143--171.
YEAR: 2002
(MASIE)Subsection : 4.2 and 1.1
Abstract: We study systems formed of 2N point vortices on a sphere with N vortices of strength +1 and N vortices of strength -1.
In this case, the Hamiltonian is conserved by the symmetry which exchanges the positive vortices with the negative vortices. We prove the existence of some fixed and relative equilibria, and then study their stability with the ``Energy Momentum Method''. Most of the results obtained are nonlinear stability results. To end, some bifurcations are described.

Author(s): Marsden, J.E., G. Misiolek, M. Perlmutter, J.-P. Ortega and T.S. Ratiu
Title: Symplectic reduction by stages
Preprint:
YEAR: 2002
(MASIE)Section: 1

Author(s): S. Mayer, J. Scheurle and S. Walcher
Title: Normal form computations for vector fields with general linearization
Preprint
YEAR: 2002
(MASIE)Section: 1

Author(s): B. Moore and S. Reich
Title: Multi-symplectic Integration Methods for Hamiltonian PDEs
Journal: Submitted
YEAR: 2002
(MASIE)Subsection : 2.3

Author(s): B. Moore and S. Reich
Title: Backward error analysis for multi-symplectic integration methods.
Journal: Submitted
YEAR: 2002
(MASIE)Subsection : 2.3

Author(s): Odzijewicz, A. and T.S. Ratiu
Title: Banach Lie-Poisson spaces and reduction
Preprint:
YEAR: 2002
(MASIE)Section : 1

Author(s): W.M. Oliva
Title: Geometric Mechanics.
Book: Lect. Notes in Math, Springer-Verlag (to appear)
YEAR: 2002
(MASIE)Section : 1

Author(s): W.M. Oliva
Title: Morse-Smale semiflows. Openess and A-stability
Journal: Fields Inst. Comm. 31, AMS, 285-307
YEAR: 2002
(MASIE)Section : Related publication

Author(s): Ortega, J.-P.
Title: The symplectic reduced spaces of a Poisson action.
Journal: C. R. Acad. Sci. Paris Sér. I Math., 334, 999--1004.
YEAR: 2002
(MASIE)Section : 1
Notes: Available at the ArXiv.

Author(s): Ortega, J.-P.
Title: Singular dual pairs.
Journal: To appear in Differential Geometry and Applications
YEAR: 2002
(MASIE)Section : 1
Notes: Available at the ArXiv.

Author(s): Ortega, J.-P.
Title: Optimal reduction.
Preprint: Available at the ArXiv.
YEAR: 2002
(MASIE)Section : 1

Author(s): Ortega, J.-P. and Ratiu, T. S.
Title: A symplectic slice theorem.
Journal: Lett. Math. Phys. Physics, 59:81--93.
YEAR: 2002
(MASIE)Section : 1
Notes: Available at the ArXiv.

Author(s): Ortega, J.-P. and T.S. Ratiu
Title: The optimal momentum map
Journal: Geometry, Dynamics, and Mechanics: 60th Birthday Volume for J.E. Marsden}, pages 329--362. P. Holmes, P. Newton, and A. Weinstein, eds., Springer-Verlag.
YEAR: 2002
(MASIE)Section : 1
Notes: Available at the ArXiv.

Author(s): G. Patrick, R.M. Roberts and C. Wulff
Title: Stability of Poisson equilibria and Hamiltonian relative equilibria by energy methods.
Preprint:
YEAR: 2002
(MASIE)Subsection : 1.1
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Author(s): R.M. Roberts, C. Wulff and J.S.W. Lamb.
Title: Hamiltonian systems near relative equilibria.
Journal: J. Differential Equations 179, 562-604.
YEAR: 2002
(MASIE)Subsection : 1.1
Abstract: We give explicit differential equations for the dynamics of Hamiltonian systems near relative equilibria. These split the dynamics into motion along the group orbit and motion inside a slice transversal to the group orbit. The form of the differential equations that
is inherited from the symplectic structure and symmetry properties of the Hamiltonian system is analysed and the effects of time reversing symmetries are included. The results will be applicable to the stability and bifurcation theories of relative equilibria of Hamiltonian systems.
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Author(s): M. Rodríguez-Olmos and M. E. Sousa-Dias.
Title: Symmetries of relative equilibria for simple mechanical systems
Journal: In Symmetry and Perturbation Theory, SPT 2002, Abenda, S., Gaeta, G., Walcher, S. (eds.), World Scientific, 221-230
YEAR: 2002
(MASIE)Subsection : 4.4, 1.1
Abstract: For symmetric simple mechanical systems we prove that the knowledge of the configuration manifold stratification by orbit types is sufficient for determining the symmetries of the relative equilibria. As an application of the results we obtain the symmetries of the (possible) relative equilibria for the affine rigid body model.

Author(s): Tanya Schmah
Title: A Cotangent Bundle Slice Theorem
Journal: (in preparation)
YEAR: 2002
(MASIE)Subsection: 1.1

Author(s): L. Sbano.
Title: Non-trivial homotopic closed orbits in Hill's region
Journal: Warwick Preprint
YEAR: 2002
(MASIE)Section : 1

Author(s): L. Sbano.
Title: Keplerian arcs with fixed energy
Journal: Warwick Preprint
YEAR: 2002
(MASIE)Section : 1

Author(s): L. Sbano.
Title: Odd-symplectic group in first order partial differential equations
Preprint: Warwick Preprint
YEAR: 2002
(MASIE)Section : 1

Author(s): Niels Soendergaard and Gregor Tanner
Title: Wave chaos in the elastic disc
Preprint: ArXiv
YEAR: 2002
(MASIE)Subection : 3.3 and 4.3
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Author(s):   Sottocornola, N.
Title: Robust homoclinic cycles in $\Bbb R^4 $
Preprint:
YEAR: 2002
(MASIE)Section : 1

Author(s): M. E. Sousa-Dias.
Title: Pseudo-rigid bodies: A geometric Lagrangian approach
Journal: Acta Applicandae Mathematicae, 70: 209-230
YEAR: 2002
(MASIE)Subsection : 4.4
Abstract: The pseudo-rigid body model is viewed in the context of continuum mechanics and elasticity theory. A Lagrangian reduction, based on variational principles, is developed for both anisotropic and isotropic pseudo-rigid bodies. For isotropic Lagrangians the reduced equations of motion for the pseudo-rigid body are a system of two (coupled) Lax equations on $so(3)\times so(3)$ and a second order differential equation on the set of diagonal matrices with positive determinant. Several examples of pseudo-rigid bodies such as stretching bodies, spinning gas could and Riemann ellipsoids are presented.
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Author(s): H. Waalkens and H. R. Dullin.
Title: Quantum monodromy in prolate ellipsoidal billiards.
Journal: Ann. Physics, 295:81-112
YEAR: 2002
(MASIE)Subsection: 3.3
Abstract: This is the third in a series of three papers on quantum billiards with elliptic and ellipsoidal boundaries. In the present paper we show that the integrable billiard inside a prolate ellipsoid has an isolated singular point in its bifurcation diagram and, therefore, exhibits classical and quantum monodromy. We derive the monodromy matrix from the requirement of smoothness for the action variables for zero angular momentum. The smoothing procedure is illustrated in terms of energy surfaces in action space including the corresponding smooth frequency map. The spectrum of the quantum billiard is computed numerically and the expected change in the basis of the lattice of quantum states is found. The monodromy is already present in the corresponding two-dimensional billiard map. However, the full three degrees of freedom billiard is considered as the system of greater relevance to physics. Therefore, the monodromy is discussed as a truly three-dimensional effect.
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Author(s): C. Wulff and R.M. Roberts
Title: Hamiltonian systems near relative periodic orbits.
Journal: SIAM J. Appl. Dyn. Sys. 1, 1-43
YEAR: 2002
(MASIE)Subsection : 1.1
Abstract: We give explicit differential equations for a symmetric Hamiltonian vector field near a relative periodic orbit. These decompose the dynamics into periodically forced motion in a Poincaré section transversal to the relative periodic orbit, which in turn forces motion
along the group orbit. The structure of the differential equations inherited from the symplectic structure and symmetry properties
of the Hamiltonian system is described and the effects of time reversing symmetries are included. Our analysis yields new results on the stability and persistence of Hamiltonian relative periodic orbits and provides the foundations for a bifurcation theory. The results are applied to a finite dimensional model for the dynamics of a deformable body in an ideal irrotational fluid.
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Author(s): C. Wulff
Title: Spiral Waves and Euclidean Symmetries.
Journal: Zeitschrift f\"ur physikalische Chemie, 216, 535-550.
YEAR: 2002
(MASIE)Subsection : Related publication
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2001

Authors: A.L. Afendikov & T.J. Bridges
Title:   Instability of the Hocking-Stewartson pulse and its implications for three-dimensional Poiseuille flow
Journal: Proc. Royal Society London A 457: 257-272
Year:    2001
(MASIE) Subsection: related publication

Authors: P. Ashwin, I. Melbourne and M. Nicol
Title:    Hypermeander of spirals; local bifurcations and statistical properties
Journal: Physica D, 156, 364--382
YEAR: 2001
(MASIE)Subsection : 1.1
Abstract: In both experimental studies and numerical simulations of waves in excitable media, rigidly rotating spiral waves are observed to undergo transitions to complicated spatial dynamics with long-term Brownian-like motion of the spiral tip. This phenomenon is
known as hypermeander.
In this paper, we review a number of recent results on dynamics with noncompact group symmetries and make the case that hypermeander may occur at a codimension two bifurcation from a rigidly rotating spiral wave. Our predictions are based on center bundle reduction (Sandstede, Scheel & Wulff), and on central limit theorems and invariance principles for noncompact group extensions of hyperbolic dynamical systems (Field, Melbourne & T\"or\"ok). These predictions are confirmed by numerical simulations of the
center bundle equations.
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Authors: F.R. Austin & T.J. Bridges
Title:   A bundle view of boundary value problems: generalizing the Gardner-Jones bundle.
Journal: Preprint
Year:    2001
(MASIE) Subsection: related publication

Author(s): M.V. Bartuccelli, G. Gentile and K. Georgiou
Title: On the dynamics of a vertically-driven damped planar pendulum.
Journal: Proceedings of the Royal Society of London, Series A, 457, 1-16
YEAR: 2001
(MASIE)Subsection : 1.3
Abstract: Results on the dynamics of the planar pendulum with parametric vertical time-periodic forcing are reviewed and extended. Numerical methods are employed to study the various dynamical features of the system about its equilibrium positions. Furthermore the dynamics of the system far from its equilibrium points is systematically investigated by using phase portraits and Poincar\'{e} sections.
The attractors and the associated basins of attraction are computed. We also calculate the Lyapunov exponents to show that for some parameter values the dynamics of the pendulum shows sensitivity to initial conditions.

Author(s):   G. Benettin, A.M. Cherubini and F. Fassò
Title: A "changing chart" symplectic algorithm for rigid bodies and other Hamiltonian systems on manifolds
Journal: SIAM Journal on Scientific Computing 23, 1189-1203
YEAR: 2001
(MASIE)Subsection: 2.2
Abstract: We revive the elementary idea of constructing symplectic integrators for Hamiltonian flows on manifolds by covering the
manifold with the charts of an atlas, implementing the algorithm in each chart (thus using coordinates), and switching among the charts whenever a coordinate singularity is approached. We show that this program can be implemented successfully by using a splitting algorithm, if the Hamiltonian is the sum $H_1+H_2$ of two (or more) integrable Hamiltonians. Profiting of integrability, we compute exactly the flows of $H_1$ and $H_2$ in each chart and thus compute the splitting algorithm on the manifold by means of its representative in any
chart. This produces a symplectic algorithm on the manifold which possesses an interpolating Hamiltonian, and hence has excellent properties of conservation of energy. We exemplify the method for a point constrained to the sphere and for a symmetric rigid body under the influence of positional potential forces.

Authors: T.J. Bridges
Title:   Transverse instability of solitary-wave states of the water-wave problem
Journal: J. Fluid Mechanics 439: 255-278
YEAR:    2001
(MASIE) Subsection: 4.5

Authors: T.J. Bridges & S. Reich
Title:   Computing Lyapunov exponents on a Stiefel manifold
Journal: PhysicaD 156: 219-238
Year:    2001
(MASIE) Subsection: related publication
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Authors: T.J. Bridges, P.E. Hydon & S. Reich
Title:   Vorticity and symplecticity in Lagrangian fluid dynamics
Journal: Preprint
Year:    2001
(MASIE) Subsection: 2.3

Authors: T.J. Bridges & K.V. Georgiou
Title:   A transverse spinning double pendulum
Journal: Chaos, Solitons & Fractals 12: 131-144
Year:    2001
(MASIE) Subsection: 1.2
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Authors: T.J. Bridges & F.E. Laine-Pearson
Title:   Multi-symplectic relative equilibria, multi-phase wavetrains and coupled NLS equations,
Journal: Studies in Applied Mathematics 107: 137-155
Year:    2001
(MASIE) Subsection 4.5

Author(s): Th.J. Bridges and S. Reich
Title: Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity
Journal: Physics Letters A 284: 184-193
YEAR: 2001
(MASIE)Subsection : 2.3
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Author(s): Th.J. Bridges and S. Reich
Title: Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow water equations.
Journal: PhysicaD 152: 491-504
YEAR: 2001
(MASIE)Subsection : 2.3
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Author(s): T.J. Bridges and G. Derks
Title: The symplectic Evans matrix, and the instability of solitary waves and fronts with symmetry.
Journal: Arch. Rat. Mech. Anal. 156, pp 1-87, 2001
YEAR: 2001
(MASIE)Subsection : 4.5
Abstract: Hamiltonian evolution equations which are equivariant with respect to the action of a Lie group are models for physical phenomena such as oceanographic flows, optical fibres and atmospheric flows, and such systems often have a wide variety of solitary wave or front solutions. In this paper, we present a new symplectic framework for analyzing the spectral problem associated with the linearization about such solitary waves and fronts. At the heart of the analysis is a multi-symplectic formulation of Hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions, with a third symplectic structure - with two-form denoted by $\Omega$ - associated with a coordinate frame moving at the speed of the wave. This leads to a geometric decomposition and symplectification of the Evans function formulation for the linearization about solitary waves and fronts.
We introduce the concept of the symplectic Evans matrix, a matrix consisting of restricted $\Omega$-symplectic forms. By applying Hodge duality to the exterior algebra formulation of the Evans function, we find that the zeros of the Evans function correspond to zeros of the determinant of the symplectic Evans matrix. Based on this formulation, we prove several new properties of the Evans function.   Restricting the spectral parameter $\lambda$ to the real axis, we obtain rigorous results on the derivatives of the Evans function near
the origin, based solely on the abstract geometry of the equations, and results for the large $|\lambda|$ behaviour which use primarily the symplectic structure, but also extend to the non-symplectic case.
The Lie group symmetry affects the Evans function by generating zero eigenvalues of large multiplicity in the so-called systems at infinity. We present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation.
By combining all these results, a new rigorous sufficient condition for instability of solitary waves and fronts is obtained.
The theory applies to a large class of solitary waves and fronts including waves which are biasymptotic to a nonconstant manifold of states as $|x|$ tends to infinity. To illustrate the theory, it is applied to three examples: a Boussinesq model from oceanography, a class of nonlinear Schrodinger equations from optics and a nonlinear Klein-Gordon equation from atmospheric dynamics.
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Author(s): T.J. Bridges and G. Derks
Title: Dimension breaking of nonlinear elliptic PDEs: satisfying the spectral condition geometrically
Preprint: UNIS preprint
YEAR: 2001
(MASIE)Subsection : 4.5
Abstract: Dimension breaking occurs when the solution of a nonlinear partial differential equation (PDE) depending on $n$ independent variables bifurcates to one depending on $n+1$. A central hypothesis in the theory of dimension breaking is that a certain operator should have a non-zero purely imaginary eigenvalue. This hypothesis is difficult to verify in general. We present a geometric theory for verifying this hypothesis. Moreover, for a large class of partial differential equations, namely multi-symplectic Hamiltonian PDEs,
we show that the verification of this hypothesis is encoded in the basic state. The theory is demonstrated by obtaining new results on dimension breaking of localized states for three examples: the (2+1)-Boussinesq equation, the Zakharov-Kuznetsov equation and the Kadomtsev-Petviashvili equation.
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Author(s): T.J. Bridges and G. Derks
Title: The symplectic Evans matrix and solitary wave instability
Journal: Conference proceedings Symmetry and Perturbation Theory 2001, editors: D. Bambusi, G. Gaeta and M. Cadoni, World
Scientific, pp 32-38.
YEAR: 2001
(MASIE)Subsection : 4.5
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Author(s): H.W. Broer and R. Roussarie
Title: Exponential confinement of chaos in the bifurcation set of real analytic diffeomorphisms.
Journal : In H.W. Broer, B. Krauskopf and G. Vegter (Eds.), Global Analysis of Dynamical Systems, Festschrift dedicated to Floris Takens for his 60th birthday. Bristol and Philadelphia IOP. ISBN 0 7503 0803 6.
YEAR: 2001
(MASIE)Subsection : 1.2 and 1.4

Author(s): H.W. Broer
Title: Coupled Hopf--bifurcations: Persistent examples of $n$--quasiperiodicity given by families of 3--jets
Journal : To appear Astérisque
YEAR: 2001
(MASIE)Subsection : 1.2

Editor(s): H.W. Broer, B. Krauskopf and G. Vegter
Title: Global Analysis of Dynamical Systems, Festschrift dedicated to Floris Takens for his 60th birthday Bristol and Philadelphia IOP. ISBN 0 7503 0803 6.
YEAR: 2001
(MASIE)Subsection : Related publication

Author(s): F. Beukers and R.Cushman.
Title: The complex spherical pendulum.
Preprint: University of Utrecht
YEAR: 2001
(MASIE)Subsection : 1.2
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Author(s): Castrillón López, M., J. Muñoz Masqué, and T.S. Ratiu
Title: Trivial Lagrangians on connections and invariance under automorphisms.
Journal: Steps in Differential Geometry, Debrecen, 2000, 77--83, Inst. Math. Inform., Debrecen, 2001.
YEAR: 2001
(MASIE)Section : 4

Author(s): Castrillón López, M., P.L. García Pérez,and T.S. Ratiu
Title: Euler-Poincaré reduction on principal bundles.
Journal: Lett. Math. Phys., 58, 167--180.
YEAR: 2001
(MASIE)Section : 4

Author(s): Castro, H. M. A.; Kobayashi, M. H.; Oliva, W. M.
Title: Partially hyperbolic $\Sigma$-geodesic flows.
Journal: Special issue in celebration of Jack K. Hale's 70th birthday, Part 3 (Atlanta, GA/Lisbon, 1998). J. Differential Equations
169, no. 1, 142--168.
YEAR: 2001
(MASIE)Subsection : Related Publication

Author(s): Cendra, H., J.E. Marsden, and T.S. Ratiu
Title: Lagrangian reduction by stages
Journal: Memoirs of the AMS, 152
YEAR: 2001
(MASIE)Section : 1 and 4

Author(s): D R J Chillingworth, E Vicente Alonso and A A Wheeler
Title: Geometry and dynamics of a nematic liquid crystal in a uniform shear flow
Journal: J. Physics A
YEAR: 2001
(MASIE)Section: 1

Author(s): Chossat, P.
Title: The bifurcation of heteroclinic cycles in systems of hydrodynamical type.
Journal: Dynamics of Continuous, Discrete, and Impulsive systems}. Special issue on Computation of normal forms
and applications.8a (4).
YEAR: 2001
(MASIE)Section: 1

Author(s): R. Cushman and J. \'{S}niatycki
Title: Differential structure of orbit spaces
Journal: Canadian J. Math., vol. 53, 715--755
Year: 2001.
(MASIE)Section: 1
Abstract: We present a new approach to singular reduction of Hamiltonian systems with symmetries. The tools we use are the category
of differential spaces of Sikorski and the Stefan-Sussmann theorem. The former is applies to analyze the diffential structure of the spaces involved and the latter is used to prove that some of these spaces are smooth manifolds. Our main result is the identification of accessible sets of the generalized distribution spanned by the Hamiltonian vector fields of invariant functions with strata of a singular reduced space. We are also able to describe the differential structure of a singular reduced space corresponding to a coadjoint orbit which need not be locally closed.

Author(s):R. Cushman and J.J. Duistermaat
Title: Non-Hamiltonian monodromy
Journal: J. Diff. Eqns., vol 172, 42-58
Year: 2001
(MASIE)Subsection:1.5

Author(s): Holger R. Dullin, Georg A. Gottwald, Darryl D. Holm
Title: An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion
Journal: Phys. Rev. Lett., 87:4501,
YEAR: 2001
(MASIE)Subsection : 4.1
Abstract: We use asymptotic analysis and a near-identity normal form transformation from water wave theory to derive a 1+1 unidirectional nonlinear wave equation that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation. This equation is one order more accurate in asymptotic approximation beyond KdV, yet it still preserves complete integrability via the inverse scattering transform (IST) method. Its traveling wave solutions contain both the KdV solitons and the CH peakons as limiting cases.
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Author(s): H. R. Dullin, P. H. Richter, A. P. Veselov, and H. Waalkens
Title: Actions of the Neumann system via Picard-Fuchs equations.
Journal: Physica D, 155, 159-183
YEAR: 2001
(MASIE)Subsection: 1.2
Abstract: The Neumann system describing the motion of a particle on an n-dimensional sphere with an anisotropic harmonic potential
has been celebrated as one of the best understood integrable systems of classical mechanics. The present paper adds a detailed discussion and the determination of its action integrals, using differential equations rather than standard integral formulas. We show that the actions of the Neumann system satisfy a Picard-Fuchs equation which in suitable coordinates has a rather simple form for arbitrary n. We also present an explicit form of the related Gauß-Manin equations. These formulas are usedfor the numerical calculation of the actions of the Neumann system.
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Author(s): Diacu, F. and T.S. Ratiu
Title: Haretu and the stability of the solar system
Journal: Romanian Astronomical Journal, 11, 85-92.
YEAR: 2001
(MASIE)Subsection: Related Publication

Author(s): H. R. Dullin and A. B¨acker
Title: About ergodicity in the family of limacon billiards.
Journal: Nonlinearity, 14:1673-1687, 2001
YEAR: 2001
(MASIE)Subsection : Related Publ.
Abstract: By continuation from the hyperbolic limit of the cardioid billiard we show that there is an abundance of bifurcations in the family of lima¸con billiards. The statistics of these bifurcation shows that the size of the stable intervals decreases with approximately the same rate as their number increases with the period. In particular, we give numerical evidence that arbitrarily close to the cardioid there are elliptic islands due to orbits created in saddle-node bifurcations. This shows explicitly that if in this one-parameter family of maps ergodicity occurs for more than one parameter, the set of these parameter values has a complicated structure.
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Author(s): El, G.A., Grimshaw, R.H.J. and Pavlov, M.V.
Title: Integrable shallow water equations and undular bores.
Journal: Stud. Appl. Math., 107, 157-186.
YEAR: 2001
(MASIE)Subsection : 4.1

Author(s): El, G.A.and Grimshaw, R.H.J.
Title: An integrable model for undular bores on shallow water.
Journal: Proceedings of the IUTAM Symposium on Feee Surface Flows, Birmingham, 2000, ed. A.C. King and Y.D. Shikhmurzaev, Kluwer, Dordrecht, 99-106.
YEAR: 2001
(MASIE)Subsection : 4

Author(s):   F. Fassò and D. Lewis
Title: Stability properties of the Riemann ellipsoids
Journal: Archive for Rational Mechanics and Analysis 158, 259-292
YEAR: 2001
(MASIE)Subsection: 4.4
Abstract: The Riemann ellipsoids are steady motions of an ideal, incompressible, self--gravitating fluid that retain an ellipsoidal
shape. The existence, properties, and stability of these steady motions have been investigated since Newton's time. Most of the
stability results are due to Riemann, who studied Lyapunov stability, and to Chandrasekhar, who studied (primarily numerically) spectral
stability, thus obtaining Lyapunov instability results. This article addresses the ``Nekhoroshev stability" (stability for finite, but very
long time scales) of those Riemann ellipsoids that are spectrally stable but of unknown Lyapunov stability. We base our analysis on a
Hamiltonian formulation of the problem derived from Riemann's original formulation (which we interpret here as a formulation on a
covering space) using recent results from Hamiltonian perturbation theory. Given the complexity of the system, we resort to numerical
calculations at certain steps of the stability analysis. As a prerequisite to our analysis, we repeat the Lyapunov and spectral stability analyses, finding important discrepancies with Chandrasekhar's findings. We provide numerical evidence that (i) There are spectrally stable ellipsoids of type II and the region of spectral stability of the ellipsoids of type III is significantly larger than that found by Chandrasekhar. The regions of spectral stability of the ellipsoids of types I, II and III have a finer and subtler structure than was previously believed. (ii) All Riemann ellipsoids, except a finite number of codimension one resonant subfamilies, are Nekhoroshev--stable.

Author(s):    F. Faure and B. Zhilinskií
Title: Qualitative features of intra-molecular dynamics. What can be learned from symmetry and topology.
Preprint: Tutorial paper for SPT 2001
YEAR: 2001
(MASIE)Subsection : 3.1 and 3.2
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Author(s):    F. Faure and B. Zhilinskií
Title: Topological properties of the Born-Oppenheimer approximation and implications for the exact spectrum.
Journal: Let. Math. Phys.
YEAR: 2001
(MASIE)Subsection : 3.1
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Author(s):    F. Faure and B. Zhilinskií
Title: Symmetry, Invariants, and Topology. V. The ring of invariant real functions on the Brillouin zone.
Journal: Phys. Rep., 341, 337-376
YEAR: 2001
(MASIE)Subsection : 3.1
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Author(s): L. Fichmann and W.M. Oliva
Title: Collision of global orbits in C-infinity retarded functional differential equations
Journal: Fields Inst. Comm. 29, AMS, 105-112
YEAR: 2001
(MASIE)Subsection : Related Publication

Author(s): L. Fichmann and W.M. Oliva
Title: One-to-oneness and hyperbolicity
Journal: Fields Inst. Comm. 29, AMS, 113-131
YEAR: 2001
(MASIE)Subsection : Related Publication

Author(s): J. Frank and S. Reich
Title: A Particle-Mesh Method for the Shallow Water Equations near Geostrophic Balance
Preprint:
YEAR: 2001
(MASIE)Subsection : 2.3
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Author(s): Grimshaw, R.
Title: Models for instability in inviscid fluid flows due to resonance between two waves.
In Book: Nonlinear Stability Analysis, Vol II, ed. L. Debnath, Advances in Fluid Mechanics, {\bf 28}, WIT Press, Southampton,
Chapter 1, 1-14.
YEAR: 2001
(MASIE)Subsection : 4

Author(s): Grimshaw, R. and Gottwald, G.
Title: Models for instability in geophysical flows.
Journal: Proceedings of the IUTAM Symposium on Advances in the Mathematical Modelling of Atmosphere and Ocean Dynamics, Limerick, 2000, ed. P.F. Hodnett, Kluwer, Dordrecht, 153-160.
YEAR: 2001
(MASIE)Subsection : 4

Author(s): Grimshaw, R., Pelinovsky, D., Pelinovsky, E. and Talipova, T.
Title: Wave group dynamics in weakly nonlinear long-wave models.
Journal: Physica D, 159, 35-57.
YEAR: 2001
(MASIE)Subsection : 4.5

Author(s): Grimshaw, R.H.J., Kuznetsov, E.A., and Shapiro, E.G.
Title: The two-parameter soliton family for the interaction of a fundamental and its second harmonic.
Journal: Physica D, 152-153, 325-339.
YEAR: 2001
(MASIE)Subsection : 4.5

Author(s): Grimshaw, R. and Christodoulides, P.
Title: Short-wave instability in a three-layered stratified shear flow.
Journal: Q.J. Mech. Appl. Math., 54, 375-388
YEAR: 2001
(MASIE)Subsection : 4.1

Author(s):   L. Grondin, D. Sadovskii and B. Zhilinskií
Title: Monodromy in systems with coupled angular momenta and rearrangement of bands in quantum spectra.
Preprint:
YEAR: 2001
(MASIE)Subsection : 3.1
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Author(s): H. Hanßmann
Title: A Survey on Bifurcations of Invariant Tori
Preprint: Princeton University
YEAR: 2001
(MASIE)Subsection : 1.1 and 1.2
Abstract: Invariant tori of dynamical systems occur both in the dissipative and in the conservative context. I focus here
on the latter, where the tori are intrinsically parametrised by the actions $y_1, ..., y_n$ conjugate to the angles $x_1, ..., x_n$ on the torus. The distribution of maximal tori in a nearly integrable Hamiltonian system is governed by the invariant tori of co-dimension one. The different Cantor families of maximal tori shrink down to normally elliptic tori and are separated by the web formed by stable and unstable manifolds of normally hyperbolic tori. The lower dimensional invariant tori form Cantor families themselves, and occurring bifurcations in turn organize the distribution of normally elliptic and hyperbolic tori.
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Author(s): H. Hanßmann and B. Sommer
Title: A degenerate bifurcation in the Hénon-Heiles family
Journal: Cel. Mech. & Dyn. Astr. 81(3), p. 249 -- 261
YEAR: 2001
(MASIE)Subsection : 1.1 and 1.2
Abstract: The normalised Hénon-Heiles family exhibits a degenerate bifurcation when passing through the separable case ``$\beta = 0$''. We clarify the relation between this degeneracy and the integrability at $\beta = 0$. Furthermore we show that the degenerate bifurcation carries over to the Hénon-Heiles family itself.

Author(s): H. Hanßmann and P. Holmes
Title: On the global dynamics of Kirchhoff's equations : Rigid body models for underwater vehicles
Journal: p. 353 -- 371 in Global Analysis of Dynamical Systems, Leiden 2001 (eds. H.W. Broer, B. Krauskopf, G. Vegter) IoP publishing
YEAR: 2001
(MASIE)Subsection : 1.1
Abstract: We study the Kirchhoff model for the motion of a rigid body submerged in an incompressible, irrotational, inviscid fluid in the absence of gravitational forces and torques. Symmetries allow reduction to a two degree-of-freedom Hamiltonian system. In [7] the existence and stability of pure and mixed mode equilibria was studied and, in [7] \S 5.2, the system was averaged, allowing further
reduction to one degree of freedom. We give an interpretation of the averaged Hamiltonian function as a normal form of order one. Iterating the process we obtain the normal form of order two, thus resolving a degeneracy noted in [7], and allowing us to prove that the (integrable) normal form of order two has heteroclinic orbits between `pure 2' and between the `pure 3' modes in a range of parameter values, and, at a critical (bifurcation) value, heteroclinic cycles linking pure 2 and pure 3 modes. We discuss the implications for the original system and the full rigid body motions.
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Author(s): Hooper, A.P. and Grimshaw, R.
Title: Transient linear growth and nonlinear effects.
Journal: Stud. Appl. Math., 106, 47-68.
YEAR: 2001
(MASIE)Subsection : 4

Author(s): H.-P. Kruse, J. Scheurle, W. Du
Title: A Two-Dimensional Version of the Camassa-Holm Equation
Journal: Proceedings of the International Conference SPT 2001, Symmetry and Perturbation Theory, Cala Gonone, Sardinia, Italy, 6-13 May 2001, eds. D. Bambusi, G. Gaeta, M. Cadoni, pp 120-127, World Sci. Publishing, River Edge, NJ.
YEAR: 2001
(MASIE)Subsection : 4.1
Abstract: The Camassa-Holm equation of shallow water theory is generalized to fluid flow over a two-dimensional sea bottom. As in the one-dimensional case the resulting equations are of Euler-Poincar\'{e} type. The dynamics of the one-dimensional Camassa-Holm equation embeds into the dynamics of the new equations.

Author(s): I.N. Kozin and R.M. Roberts.
Title: Monodromy in the spectrum of an axisymmetric top molecules in an electric field.
Preprint:
YEAR: 2001
(MASIE)Subsection : 2.1 and 2.3

Author(s): I. Kupka and W.M. Oliva
Title: The Non-Holonomics Mechanics.
Journal: Special issue in celebration of Jack K. Hale's 70th birthday, Part 3 (Atlanta, GA/Lisbon, 1998). J. Differential Equations
169, no. 1, 169--189.
YEAR: 2001
(MASIE)Subsection : 1.5 and Related publication
Abstract: The approaches for the study of mechanical systems with non holonomic constraints are presented: d'Alembertian mechanics and variational (vaconomic) mechanics. The first is equivalent to d'Alembert principle and the second comes from a variational principle .Corresponding to the two approaches, d'Alembertian and vaconomic trajectories are introduced. The version of the classical Liouville theorem for the conservation of volume is proved in the context of d'Alembertian mechanics. A caracterization for the notion of regular and singular trajectories is presented. The flow corresponding to the regular vaconomic trajectories is Hamiltonian.

Authors: J. S. W. Lamb, I. Melbourne and C. Wulff.
Title:    Bifurcation from periodic solutions with spatiotemporal symmetry, including resonances and mode interactions.
Preprint
YEAR: 2001
(MASIE)Subsection : 1.1
Abstract: We study local bifurcation in equivariant dynamical systems from periodic solutions with a mixture of spatial and spatiotemporal symmetries.
In previous work, we focused primarily on codimension one bifurcations. In this paper, we show that the techniques used in the codimension one analysis can be extended to understand also higher codimension bifurcations, including resonant bifurcations and mode interactions. In particular, we present a general reduction scheme by which we relate bifurcations from periodic solutions to bifurcations from fixed points of twisted equivariant diffeomorphisms, which in turn are linked via normal form theory to bifurcations from equilibria of equivariant vector fields.
We also obtain a general theory for bifurcation from relative periodic solutions and we show how to incorporate time-reversal symmetries into our framework.
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Author(s): C. Wulff , J. Lamb and I. Melbourne
Title: Bifurcation from relative periodic solutions
Journal: Erg. Th. Dyn. Syst., 21, 1--31
YEAR: 2001
(MASIE)Subsection : 1.1
Abstract: Relative periodic solutions are ubiquitous in dynamical systems withcontinuous symmetry. Recently, Sandstede, Scheel and Wulff derived a center bundle theorem, reducing local bifurcation from relative periodic solutions to a finite dimensional problem. Independently, Lamb and Melbourne showed how to systematically
study local bifurcation from isolated periodic solutions with discrete spatiotemporal symmetries.
In this paper, we show how the center bundle theorem, when combined with certain group theoretic results, reduces bifurcation from relative periodic solutions to bifurcation from isolated periodic solutions. In this way, we obtain a systematic approach to the study of local bifurcation from relative periodic solutions.
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Author(s): J. Lamb, I. Melbourne and C. Wulff
Title: General bifurcation from periodic solutions with spatio-temporal symmetries, including mode interactions and resonances.
Journal: Preprint
YEAR: 2001
(MASIE)Section : 1
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Author(s): B. Leimkuhler and S. Reich
Title: A reversible averaging integrator for multiple time-scale dynamcis.
Journal: J. Comput. Physics 171, 95-114
YEAR: 2001
(MASIE)Subsection : 2.1

Author(s): Chjan Lim, James Montaldi and Mark Roberts
Title: Relative equilibria of point vortices on the sphere.
Jounal: Physica D 148, 97-135.
YEAR: 2001
(MASIE)Subsection : 1.1 and 4.2
Abstract: We prove the existence of many different symmetry types of relative equilibria for systems of identical point vortices on a non-rotating sphere. The proofs use the rotational symmetry group SO(3) and the resulting conservation laws, the time-reversing reflectional symmetries in O(3), and the finite symmetry group of permutations of identical vortices. Results include both global existence theorems and local results on bifurcations from equilibria. A more detailed study is made of relative equilibria which consist of two parallel rings with $n$ vortices in each rotating about a common axis. The paper ends with discussions of the bifurcation diagrams for systems of 3, 4, 5 and 6 identical vortices.
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Author(s):   L. Michel and   B. I. Zhilinskií
Title: Symmetry, Invariants, and Topology. III. Rydberg states of atoms and molecules. Basic group theoretical and topological analysis.
Journal: Phys. Rep.,   341, 175-264
YEAR: 2001
(MASIE)Subsection : 3.2 and 3.3
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Author(s):   L. Michel and   B. I. Zhilinskií
Title: Symmetry, Invariants, and Topology. I. Basic Tools.
Journal: Phys. Rep.,   341, 11-84
YEAR: 2001
(MASIE)Subsection : 3.1
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Author(s): W.M. Oliva
Title: Mechanics on Riemannian Manifolds
Journal: Progr. Nonlinear Differential Equations Appl. 43, Birkhauser, 65-84.
YEAR: 2001
(MASIE)Section : 1

Author(s): Ortega, J.-P. and T.S. Ratiu
Title: Critical point theory and Hamiltonian dynamics around critical elements
Journal: Symmetry and Perturbation Theory (Proceedings of the international conference SPT2001, Cala Gonone), D. Bambusi, M. Cadoni and G. Gaeta editors, World Scientific, Singapore, 151--158.
YEAR: 2001
(MASIE)Subsection : 1.1

Author(s): Ortega, J.-P. and T.S. Ratiu
Title: A symplectic slice theorem
Journal: Lett. Math. Phys., 59, 81--93.
YEAR: 2001
(MASIE)Section : 4

Author(s): Ortega, J.-P. and T.S. Ratiu
Title: The dynamics around stable and unstable Hamiltonian relative equilibria
Preprint:
YEAR: 2001
(MASIE)Subection : 1.1

Author(s): G. Patrick, R.M. Roberts and C. Wulff
Title: Stability of Hamiltonian relative equilibria by energy methods.
Journal: Symmetry and Perturbation Theory: Proceedings of the International Conference SPT2001 (eds: D. Bambusi, M. Cadoni and G. Gaeta), World Scientific, Singapore, 2001.
YEAR: 2001
(MASIE)Subection : 1.1
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Author(s): U. Scheerer, C. Wulff
Title: Reduced dynamics for momentum maps with cocycle
Journal: C. R. Acad. Sc., Série I, Math., 333, 99-104
YEAR: 2001
(MASIE)Section : 1
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Author(s): Troy R. Smith, H. Hanßmann and Naomi Ehrich Leonard
Title: Orientation Control of Multiple Underwater Vehicles with Symmetry-Breaking Potentials
Journal: p. 4598 -- 4603 in Proceedings of the 40th IEEE Conference on Decision and Control, Orlando 2001 (eds. D.W. Repperger et al.) IEEE
YEAR: 2001
(MASIE)Subsection : Related publication
Abstract: We present a control strategy for stable orientation alignment of autonomous vehicles traveling together as a coordinated group in three-dimensional space. The control law derives from an artificial potential that depends only on the relative orientation of pairs of vehicles. The result is a controlled system of coupled rigid bodies with partially broken rotational symmetry. Semidirect
product reduction theory is used to study the closed-loop dynamics, and the energy-Casimir method is applied to the reduced
dynamics to prove stability of an alignment of vehicles translating in parallel along the same body axis. For clarity, the
theory is described in detail for the case of two underwater vehicles, and the extension to an arbitrary number of underwater vehicles is summarized.

Author(s):   Sottocornola, N.
Title: Sur la classification des cycles homoclines.
Journal: C. R. Acad. Sci. Paris S\'er. I Math., 332, 695--698.
YEAR: 2001
(MASIE)Section : 1

Author(s):   Sottocornola, N.
Title: Complete classification of of homoclinic cycles in $\Bbb R^4$.
Journal: C. R. Acad. Sci. Paris S\'er. I Math., 334, 859--864.
YEAR: 2001
(MASIE)Section : 1

Author(s):    Chy. Van Hecke, D. Sadovskii, B. I. Zhilinskií, V. Boudon
Title: Rotational-vibrational relative equilibria and the structure of quantum energy spectrum of the tetrahedral molecule P$_4$.
Preprint:
YEAR: 2001
(MASIE)Subsection : 3.1, 3.2 and 3.3
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Author(s):   Vu Ngoc S.
Title: On semi-global invariants of focus-focus singularities.
Preprint: Institut Fourier 2001. (To appear in Topology)
YEAR: 2001
(MASIE)Subsection : 1.2
Abstract:   This article gives a classification, up to symplectic equivalence, of singular Lagrangian foliations given by a completely integrable system of a 4-dimensional symplectic manifold, in a full neighbourhood of a singular leaf of focus-focus type.
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Author(s):   Vu Ngoc S.
Title: Quantum Monodromy and Bohr-Sommerfeld rules
Journal: Lett. Math. Phys., vol 55, number 3, pp. 205--217
YEAR: 2001
(MASIE)Subsection : 1.2
Abstract: This article is based on a talk given in Dijon for the 2000 summer school ``Topological and Geometrical Methods: Applications to Dynamical Systems''.
The standard definitions of the monodromy invariant for completely integrable classical systems are reviewed, and the link to the quantum monodromy observed in the joint spectrum of commuting operators is explained. The mathematical treatment relies on a modern and attractive version of the Bohr-Sommerfeld quantisation rules.

Author(s): C. Wulff
Title: Persistence of relative equilibria in Hamiltonian systems with noncompact symmetries
Preprint: Freie Universitát Berlin, 2001
(MASIE)Subsection : 1.1
YEAR: 2001
Abstract:   We prove new results on the persistence of Hamiltonian relative equilibria with generic velocity-momentum pairs in the case of non-compact non-free group actions and taking into account time reversibility. Our starting point is a relative equilibrium which is
non-degenerate modulo isotropy which, in the case of a generic momentum implies persistence of the given relative equilibrium to all
nearby momentum values with the same isotropy. We show that the analysis of the persistence problem involves the study of a singular algebraic variety which is determined solely by the symmetry group of the problem. We present persistence results for relative equilibria
with velocity-momentum pairs which are regular points of this variety and give sufficient conditions for a velocity-momentum pair to be regular. We apply our results to relative equilibria of Euclidean equivariant systems, including models of rigid bodies in fluids.
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Author(s): B. I. Zhilinskií
Title: Symmetry, Invariants, and Topology. II. Symmetry, invariants, and topology in moleculsr models.
Journal: Phys. Rep.,   341, 85-171
YEAR: 2001
(MASIE)Subsection : 3.1, 3.2 and 3.3
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Author(s): S. Walcher
Title: On cooperative systems with respect to arbitrary orderings
Journal: J. Math. Analysis Appl. 263, 543-554.
YEAR: 2001
(MASIE)Section: 1

2000

Authors: T.J. Bridges
Title:   Universal geometric condition for the transverse instability of solitary waves
Journal: Physical Review Letters 84: 2614-2617
Year:    2000
(MASIE) Subsection: 4.5
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Author(s): H.W. Broer and F.O.O. Wagener
Title: Quasi--periodic stability of subfamilies of an unfolded skew Hopf bifurcation.
Journal : Archive Rat. Mech. An. 152 pp 34-43
YEAR: 2000
(MASIE)Subsection : 1.2

Author(s): H.W. Broer and C. Simó
Title: Resonance tongues in Hill's equations: a geometric approach,
Journal Journ. Diff. Eqns. 166 pp 290-327.
YEAR: 2000
(MASIE)Subsection : 1.2 and 1.3

Author(s): H.W. Broer and C. Simó
Title: Reducible linear quasi--periodic systems with positive Lyapunov exponent and varying rotation number,
Journal : Journ. Diff. Eqns. 168 pp 60-66.
YEAR: 2000
(MASIE)Subsection : 1.2

Author(s): Pascal Chossat, Juan-Pablo Ortega and Tudor S. Ratiu
Title: Hamiltonian Hopf bifurcation with symmetry
Preprint:
YEAR: 2000
(MASIE)Subsection : 1.1
Abstract: In this paper we study the appearance of branches of relative periodic orbits in Hamiltonian Hopf bifurcation processes in the presence of compact symmetry groups that do not generically exist in the dissipative framework. The theoretical study is illustrated with several examples.
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Author(s): Clarke, S., Grimshaw, R., Miller, P., Pelinovsky, E., and Talipova, T
Title: On the generation of solitons and breathers in the modified Korteweg-de Vries equation.
Journal: Chaos, 10, 383-392.
YEAR: 2000
(MASIE)Subsection : 4.1

Author(s): Clarke, S., Grimshaw, R., and Malomed, B.
Title: Soliton formation from a pulse passing the zero-dispersion point in a nonlinear Schrodinger equation.
Journal: Phys. Rev. E, 61, 5794-5801.
YEAR: 2000
(MASIE)Subsection : 4.5

Author(s): COLIN de VERDIERE, Yves and VU NGOC, San
Title: Singular Bohr-Sommerfeld rules for 2D integrable systems.
Preprint: Institut Fourier
YEAR: 2000
(MASIE)Subsection : 1.2 , 3.1 and 3.3
Abstract: In this paper, we describe Bohr-Sommerfeld rules for semi-classical completely integrable systems with 2 degrees of freedom with non degenerate singularities (Morse-Bott singularities) under the assumption that the energy level of the first Hamiltonian is non singular. The more singular case of focus-focus singularities is studied by the second author, while the case of 1 degree of freedom has been studied by the first author in collaboration with B. Parisse.
Our theory is applied to some famous examples: the geodesics of the ellipsoid, the $1:2$-resonance, and Schrödinger operators on the sphere $S^2$. A numerical test shows that the semiclassical Bohr-Sommerfeld rules match very accurately the ``purely quantum'' computations.
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Author(s):   G. Dhont, D. A. Sadovskii, B. I. Zhilinskii, and V. Boudon
Title: Analysis of the ``Unusual'' Vibrational Components of Triply Degenerate Vibrational Mode nu_6 of Mo(CO)_6 Based on the Classical Interpretation of the Effective Rotation-Vibration Hamiltonian
Journal: J. Mol. Spectrosc., 201, 95-108
YEAR: 2000
(MASIE)Subsection: 3.1 and 3.2
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Author(s): H. R. Dullin, J. D. Meiss, and D. G. Sterling.
Title: Generic twistless bifurcations.
Journal: Nonlinearity, 13. 203-224
YEAR: 2000
(MASIE)Subsection: 1.2
Abstract: We show that in the neighborhood of the tripling bifurcation of a periodic orbit of a Hamiltonian flow or of a fixed point of an area-preserving map, there is generically a bifurcation that creates a ``twistless'' torus. At this bifurcation, the twist, which is the derivative of the
rotation number with respect to the action, vanishes. The twistless torus moves outward after it is created and eventually collides with the saddle-center bifurcation that creates the period three orbits. The existence of the twistless bifurcation is responsible for the breakdown of
the nondegeneracy condition required in the proof of the KAM theorem for flows or the Moser twist theorem for maps. When the twistless torus has a rational rotation number, there are typically reconnection bifurcations of periodic orbits with that rotation number.

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Author(s):   Frederic Faure and Boris I. Zhilinskií
Title: Topological Chern indices in molecular spectra
Journal: Phys. Rev. Lett. , 85, 960-963. (quant-ph/9912091).
YEAR: 2000
(MASIE)Subsection: 3.1
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Author(s): G. Fusco and W.M. Oliva
Title: Formation of symmetric structures in the dynamics of repelling particles.
Journal: Arch. Ration. Mech. Anal. 151, no. 2, 95--123.
YEAR: 2000 (to appear)
(MASIE)Subsection : 1.2
Abstract: One considers Hamiltonian systems corresponding to the motions of a system of N repelling particles evolving in the space under the action deriving from a very long range potential energy; it is analysed the   asymptotic behaviour of the system for the case $U = - \ln r$ and $U = - \frac{r^\gamma}{\gamma} + \frac{1}{r}, 0< \gamma<1$. Only special "asymptotic shapes" are reached, which may present quite interesting symmetries and correspond to the critical points of a gradient system. The relationships between the original Hamiltonian and the asymptotic gradient system are discussed.

Authors: M. Golubitsky, V. G. LeBlanc and I. Melbourne
Title:    Hopf bifurcation from rotating waves and patterns in physical space
Journal: J. Nonlin. Sci., 10, 69--101
YEAR : 2000
(MASIE)Subsection : 1.1
Abstract: Hopf bifurcations from time periodic rotating waves to two frequency tori have been studied for a number of years by a variety of authors including Rand and Renardy. Rotating waves are solutions to partial differential equations where time evolution is the same as spatial rotation. Thus rotating waves can exist mathematically only in problems that have at least SO(2) symmetry. In this paper we study the effect on this Hopf bifurcation when the problem has more than SO(2) symmetry. These effects manifest themselves in physical space and not in phase space. We use as motivating examples the experiments of Gorman et al. on porous plug burner flames,
of Swinney et al. on the Taylor-Couette system, and of a variety of people on meandering spiral waves in the Belousov-Zhabotinsky reaction. In our analysis we recover and complete Rand's classification of modulated wavy vortices in the Taylor-Couette system.
It is both curious and intriguing that the spatial manifestations of the two frequency motions in each of these experiments is different and it is these differences that we seek to explain. In particular, we give a mathematical explanation of the differences between the nonuniform rotation of cellular flames in Gorman's experiments and the meandering of spiral waves in theBelousov-Zhabotinsky reaction.
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Author(s): Ionescu D. and Soós E.
Title: Electrogravitational Field Produced by a Charged Mass Point in RTG
Journal: Rev. Roum. Math. P. Appl. 45, No. 2, 251-260
YEAR: 2000
(MASIE)Section: 1

Author(s): Ionescu D. and Soós E.
Title: Consequences of the Causality Principle in the Relativistic Theory of Gravitation
Journal: Proc. of the XXIII Intern. Workshop of High Energy Physics and Field Theory, Protvino (Russia), 180-190.
YEAR: 2000
(MASIE)Section: 1

Author(s):   J. S. Kim, L. Michel, and B. I. Zhilinskií
Title: Invariant theory in crystal symmetry.
Journal: Proceedings ... , Poland . World Scientific
YEAR: 2000
(MASIE)Subsection : 3.1
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Author(s): James Montaldi and Mark Roberts
Title: A note on semisymplectic actions of Lie groups
Journal: C. R. Acad. Sci. Paris 330, 1079-1084.
YEAR: 2000
(MASIE)Subsection : Related publication
Abstract: A semisymplectic action of a Lie group on a symplectic manifold is one where each element of the group acts either symplectically or antisymplectically. We find conditions that a semisymplectic action descends to an action on the symplectic reduced spaces. We consider a few examples, and in particular apply these ideas to reduction of $N$-body systems with Galilean invariance.

Author(s): G. Tanner, K. Richter, and J. M. Rost
Title: The theory of two electron spectra: From the ground state to complete fragmentation
Journal: Review of Modern Physics (to appear).
YEAR: 2000
(MASIE)Subsection : 3.3

Author(s): W.M. Oliva, J. S. Santos, P.Z. Táboas
Title: A set of global bounded solutions for a Volterra system of retarded Equations on $R^3_+$ .
Journal: NoDEA, Non Linear Diff. Equ. appl., 7, 207-231, 2000
YEAR: 2000
(MASIE)Subsection : Related publication
Abstract: It is proved the existence of a compact set K, invariant under the flow of a Volterra system of retarded equations on $R^3_+$ with lag $r>0$; K is homeomorphic to a solid tri-dimensional cylinder. The boundary $\partial K$ of K is the union of a closed bi-dimensional cylinder $C(K)$ with two open disks (the two basis of the cylinder K). $C(K)$ is the union of a continuous one-parameter family of r-periodic orbits of the retarded Volterra system and any r-periodic orbit of the retarded system is contained in K. The flow, restricted to K, of the system of the retarded equations, is the flow of a $C^1$-vector field.

Author(s): Juan-Pablo Ortega
Title: Relative normal modes for nonlinear Hamiltonian systems.
Preprint:
YEAR: 2000
(MASIE)Subsection : 1.1
Abstract: An estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given. This result constitutes a generalization to the Hamiltonian symmetric framework of a classical result by Weinstein and Moser on the existence of periodic orbits in the energy levels surrounding a stable equilibrium.The estimate obtained is very precise in the sense that it provides a lower bound for the number of relative periodic orbits at each prescribed energy and momentum values neighboring the stable relative equilibrium in question and with any prefixed (spatiotemporal) isotropy subgroup. Moreover, it is easily computable in particular examples. It is interesting to see how in our result the existence of non trivial relative periodic orbits requires (generic) conditions on the higher order terms of the Taylor expansion of the Hamiltonian function, in
contrast with the purely quadratic requirements of the Weinstein--Moser Theorem, which emphasizes the highly non linear character of the relatively periodic dynamical objects.
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Author(s): Juan-Pablo Ortega, Tudor S. Ratiu
Title: The dynamics around stable and unstable Hamiltonian relative equilibria
Preprint:
YEAR: 2000
(MASIE)Subsection : 1.1
Abstract: For a symmetric Hamiltonian system on a symplectic manifold, lower bounds for the number of relative equilibria surrounding stable and formally unstable relative equilibria on nearby energy levels are given. Lower bounds for previous merely existential results of Lerman and Tokieda on relative periodic orbits are obtained as corollaries of the presented techniques, as well as an improvement of a previous result of Montaldi, Roberts, and Stewart, on the existence of periodic orbits around a symmetric Hamiltonian equilibrium, already obtained by Bartsch.
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Author(s): G.W. Patrick and R.M. Roberts.
Title: The transversal relative equilibria of a Hamiltonian system with symmetry.
Journal: Nonlinearity 13, 2089-2105.
YEAR: 2000
(MASIE)Subsection : 1.1

Author(s): VU NGOC, San
Title: Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type.
Journal: Comm. Pure and Applied Math 53(2), pp. 143--217
YEAR: 2000
(MASIE)Subsection : 1.2 / 3.3
Abstract: We present a detailed study, in the semi-classical regime $h \to 0$, of microlocal properties of systems of two commuting $h$-pseudodifferential operators $P_1(h)$, $P_2(h)$ such that the joint principal symbol $p=(p_1,p_2)$ has a special kind of singularity called a focus-focus singularity. Typical examples include the quantum
spherical pendulum or the quantum Champagne bottle.
In the spirit of Colin de Verdière and Parisse, we show that such systems have a universal behavior described by singular quantization conditions of Bohr-Sommerfeld type, involving geometrical invariants of the associated singular lagrangean foliation.
These conditions are used to give a precise description of the joint spectrum of such systems, including the phenomenon of quantum monodromy and different formulations of the counting function for the joint eigenvalues close to the singularity, in which a logarithm of the semi-classical constant $h$ appears. Thanks to numerical computations done by M.S. Child for the case of the Champagne bottle, we are able to accurately illustrate our statements.

Author(s):   B. I. Zhilinskií
Title: Rearrangement of energy bands: Quantum, semi-quantum, and classical models.
Journal: Proceedings ... , 67 RCP, Strasbourg.
YEAR: 2000
(MASIE)Subsection : 3.1 and 3.2
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Author(s):   B. I. Zhilinskií
Title: Symmetry, invariants, topology in molecules.
Journal: Proceedings of ICGTMP 2000, Dubna
YEAR: 2000
(MASIE)Subsection : 3.1 and 3.2
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Author(s):   B. I. Zhilinskií, M.I. El Idrissi, and M. Herman
Title: The vibrational energy pattern in acetylene (VI): Inter and intrapolyad structures
Journal: J.Chem. Phys. , 113, 7885-7890
YEAR: 2000
(MASIE)Subsection : 3.1 and 3.2
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1999

Authors:   P. Ashwin, I. Melbourne and M. Nicol
Title:     Drift bifurcations of relative equilibria and transitions of spiral waves
Journal:   Nonlinearity, 12, 741--755
YEAR: 1999
(MASIE)Subsection : 1.1
Abstract: We consider dynamical systems that are equivariant under a noncompact Lie group of symmetries and the drift of relative equilibria in such systems. In particular, we investigate how the drift for a parametrized family ofnormally hyperbolic relative equilibria can change character at what we call a `drift bifurcation'. To do this, we use results of Arnold to analyze parametrized families of elements in the Lie algebra of the symmetry group.
We examine effects in physical space of such drift bifurcations for planar reaction-diffusion systems and note that these effects can explain certain aspects of the transition from rigidly rotating spirals to rigidly propagating `retracting waves'. This is a bifurcation observed in numerical simulations of excitable media where the rotation rate of a family of spirals slows down and gives way to a semi-infinite translating wavefront.
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Author(s): Barth, E., Leimkuhler, B., and Reich, S.
Title: Semi-explicit, variable-stepsize integrator for constrained dynamics
Journal: SIAM J. Sci. Comp
YEAR: 1999
(MASIE)Subsection : 2.3

Author(s): L. Bates and R. Cushman
Title: What is a completely integrable nonholonomic dynamical system?
Journal: Reports on Mathematical Physics , 44, 29-35.
YEAR: 1999
(MASIE)Subsection : 1.5
Abstract: We compare the geometry of a toral fibration defined by the common level sets of the integrals of a Liouville integrable Hamiltonian system with a toral fibration coming from a completely integrable nonholonomic system. We illustrate their differences using the following examples: the nonholonomic oscillator, Chaplygin's skate, Routh's sphere and the rolling ellipsoid of revolution.

Author(s):   G. Benettin and F. Fassò
Title: From Hamiltonian perturbation theory to symplectic integrators and back
Journal: Applied Numerical Mathematics 29, 73-87 (1999).
YEAR: 1999
(MASIE)Subsection: Related publication
Abstract: Hamiltonian perturbation theory explains how symplectic integrators work and, in particular, why they can be used to measure
extremely small energy exchanges between different degrees of freedom in molecular collision problems. Conversely, numerical experiments based on symplectic integrators permit a detailed understanding of the dynamics of nearly integrable Hamiltonian systems, thus providing a valuable support to Hamiltonian perturbation theory.
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Author(s): S. Bond, B. Laird, and B. Leimkuhler
Title: The Nosé-Poincaré Method for Constant Temperature Molecular Dynamics, with
Journal: J. Comput. Phys. 151, 114-134
YEAR: 1999

Author(s): T.J. Bridges and G. Derks
Title: Unstable eigenvalues and the linearisation about solitary waves and fronts with symmetry
Journal: Proceedings of the Royal Society London, A:455, pp 2427-2469, 1999.
YEAR: 1999
(MASIE)Subsection : 4.5
Abstract: The linear stability of solitary-wave or front solutions of Hamiltonian evolutionary equations, which are equivariant with respect to a Lie group, is studied. The organizing centre for the analysis is a multi-symplectic formulation of Hamiltonian PDEs where a distinct symplectic operator is assigned for time and space. This separation of symplectic structures leads to new characterizations of the following components of the analysis. The states at infinity are characterized as manifolds of relative equilibria associated with the spatial symplectic structure. The momentum of the connecting orbit, or shape of the solitary wave, considered as a heteroclinic orbit in a phase space representation, is given a new characterization as a one-form on the tangent space to the heteroclinic manifold and this one-form is a restriction of the temporal symplectic structure. For the linear stability analysis, a new symplectic characterization of the Evan's function and its derivatives are obtained, leading to an abstract geometric proof of instability for a large class of solitary-wave states of equivariant Hamiltonian evolutionary PDEs. The theory sheds new light on several well-known models: the gKdV equation, a Boussinesq system and a nonlinear wave equation. The generalization to solitary-waves associated with multi-dimensional heteroclinic manifolds and the implications for solitary waves or fronts which are biasymptotic to invariant manifolds such as periodic states are also discussed.
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Author(s): H. Broer, I. Hoveijn, M. van Noort and G. Vegter
Title: The inverted pendulum: a singularity theory approach
Journal: Journ. Diff. Eqns. 157, 120--149 (1999).
YEAR: 1999
(MASIE)Subsection :

Author(s): H.M.A. Castro and W.M. Oliva
Title: Anosov flows induced by partially hyperbolic $\Sigma$-geodesic flows.
Journal: Resenhas IME-USP, vol.4, N. 2, 227-246.
YEAR: 1999
(MASIE)Subsection : Related publication
Abstract: We construct Anosov flows related with partially hyperbolic flows on codimension 1 non-integrable orientable distribuitions of compact Riemannian manifolds. The distributions are constant umbilical and need a volume preserved. The manifolds are supposed to have sufficiently negative sectional curvatures on the planes contained in the distribution.

Author(s): M S Child, T Weston and J Tennyson
Title: Quantum monodromy in the spectrum of H2O and other systems
Journal: Molec Physics 96, 371-379
YEAR: 1999
(MASIE)Subsection : 3.3
Abstract: The concept of quantum monodromy is introduced to give insight into the energy levels of systems with cylindrically symmetrical potential energy barriers. The $K$ structure of bending progressions of bent molecules and the pendular states of dipolar molecules in strong electric fields are taken as molecular examples. Results are given for a two dimensional champagne bottle model, for six computed bending progressions for water taken from the compilation of Partridge and Schwenke and for the quantal sherical pendulum. Sharp changes in the energy level patterns around the barrier energy are related to changes in the forms of the relevant classical trajectories. Analytical insight into the mathematical origin of the monodromy are also given.

Author(s): Pascal Chossat, Debra Lewis, Juan-Pablo Ortega and Tudor S. Ratiu
Title: Bifurcation of relative equilibria in mechanical systems with symmetry
Preprint:
YEAR: 1999
(MASIE)Subsection : 1.1
Abstract: The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and construct a collection of implicitly defined functions and reduced equations describing the set of relative equilibria in a neighborhood of a given relative equilibrium. The structure of the reduced equations is studied in a few relevant situations. In particular, a persistence result of Lerman and Singer [LS98] is generalized to the framework of Abelian proper actions. Also, a Hamiltonian version of the Equivariant Branching Lemma and a study of bifurcations with maximal isotropy are presented. An elementary example is presented to illustrate the use of this approach.
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Author(s): Cirilli, S., Hairer, E., and Leimkuhler, B
Title: Asymptotic Error Analysis of the Adaptive Verlet Method
Journal: BIT 39, 25-33, 1999
YEAR: 1999
(MASIE)Subsection : Related publication (Section 4)

Author(s): Creagh, St. and N. D. Whelan
Title: Homoclinic Structure Controls Chaotic Tunnelling
Journal: Phys. Rev. Lett. 5237, 82, (1999).
YEAR: 1999
(MASIE)Subsection : 3.3
Abstract: Tunnelling from a chaotic potential well is explained in terms of a set of complex periodic orbits which contain information about the real dynamics inside the well as well as the complex dynamics under the confining barrier. These orbits are associated with trajectories which are homoclinic to a real trajectory emerging from the optimal tunnelling path. The theory is verified by considering a model double-well problem.

Author(s): Creagh, St. and N. D. Whelan
Title: A Matrix Element for Chaotic Tunnelling Rates and Scarring Intensities
Journal: Ann. Phys. 272, 196 (1999).
YEAR: 1999
(MASIE)Subsection : 3.3
Abstract: It is shown that tunnelling splittings in ergodic double wells and resonant widths in ergodic metastable wells can be approximated as easily-calculated matrix elements involving the wavefunction in the neighbourhood of a certain real orbit. This orbit is a continuation of the complex orbit which crosses the barrier with minimum imaginary action. The matrix element is computed by integrating across the orbit in a surface of section representation, and uses only the wavefunction in the allowed region and the stability properties of the orbit. When the real orbit is periodic, the matrix element is a natural measure of the degree of scarring of the wavefunction. This scarring measure is canonically invariant and independent of the choice of surface of section, within semiclassical error. The result can alternatively be interpretated as the autocorrelation function of the state with respect to a transfer operator which quantises a certain complex surface of section mapping. The formula provides an efficient numerical method to compute tunnelling rates while avoiding the need for the exceedingly precise diagonalisation endemic to numerical tunnelling calculations.

Author(s): Creagh, St. and N. D. Whelan
Title: Regular Tunnelling Sequences in Mixed Systems
Preprint: chaodyn 9910021
YEAR: 1999
(MASIE)Subsection : 3.3
Abstract: We show that the pattern of tunnelling rates can display a vivid and regular pattern when the classical dynamics is of mixed chaotic/regular type. We consider the situation in which the dominant tunnelling route connects to a stable periodic orbit and this orbit is surrounded by a regular island which supports a number of quantum states. We derive an explicit semiclassical expression for the positions and tunnelling rates of these states by use of a complexified trace formula.

Author(s): Creagh, St. and N. D. Whelan
Title: The Statistics of Chaotic Tunnelling
Preprint: chaodyn 9909046
YEAR: 1999
(MASIE)Subsection : 3.3
Abstract: We discuss the statistics of tunnelling rates in the presence of chaotic classical dynamics. This applies to resonance widths in chaotic metastable wells and to tunnelling splittings in chaotic symmetric double wells. The theory is based on using the properties of a semiclassical tunnelling operator together with random matrix theory arguments about wave function overlaps. The resulting distribution depends on the stability of a specific tunnelling orbit and is therefore not universal. However it does reduce to the universal Porter-Thomas form as the orbit becomes very unstable. For some choices of system parameters there are systematic deviations which we explain in terms of scarring of certain real periodic orbits. The theory is tested in a model symmetric double well problem and possible experimental realisations are discussed.

Author(s): R. Cushman, S. Ferrer, H. Hanßmann
Title: Singular reduction of axially symmetric perturbations of the isotropic harmonic oscillator.
Journal: Nonlinearity ,12, 389-410
YEAR: 1999
(MASIE)Subsection : 1.2
Abstract: The normal form of an axially symmetric perturbation of the isotropic harmonic oscillator is invariant under a 2-torus action and thus integrable in three degrees of freedom. The reduction of this symmetry is performed in detail, showing how the
singularities of the reduced phase space determine the distribution of periodic orbits and invariant 2-tori in the original perturbation. To illustrate these results a particular quartic perturbation is analysed.

Author(s): R. Cushman and D.A. Sadovskii
Title: Monodromy in perturbed Kepler systems: hydrogen atom in crossed fields.
Journal: Europhysics Letters, 47, 1-7
YEAR: 1999
(MASIE)Subsection : 1.2 and 3.3
Abstract: We show that an integrable approximation to the hydrogen atom in orthogonal electric and magnetic fields found by a two step normalization procedure has monodromy. Monodromy is a fundamental property of the dynamical system which precludes the existence of globally defined action angle variables and quantum numbers.
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Author(s): R. Cushman and D.A. Sadovskii
Title: Monodromy in the hydrogen atom in crossed fields.
Preprint: 1107
YEAR: 1999
(MASIE)Subsection : 1.2 and 3.3
Abstract: A more detailed version of Europhysics Letters 47 (1999) 1-7.
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Author(s): G. Derks
Title: Families of relative equilibria in Hamiltonian systems with dissipation.
Journal: Conference proceedings Symmetry and Perturbation Theory 1998, Editors: A. Degasperis and G. Gaeta. World Scientific, 1999.
YEAR: 1999
(MASIE)Subsection : 1.3 and 4.5
Abstract: In this note the influence of dissipation on families of relative equilibria in Hamiltonian systems will be considered. Relative equilibria can be described as critical points of an appropriate functional. This characterisation can be used to give sufficient conditions such that in finite dimensional systems with dissipation the extremal families of relative equilibria are stable under dissipation. Furthermore, a full class of families of relative equilibria in the Navier-Stokes equations will be analysed. For these families it will be shown that the extremal family of relative equilibria is an attractor and the non-extremal families of relative equilibria are unstable.

Author(s): G. Dhont and D. Sadovskii and B. Zhilinskii and V. Boudon.
Title: Analysis of the ``unusual'' vibrational components of triply degenerate mode $\nu_6$ of {M}o{CO}$_6$ based on the classical interpretation of the effective rotation-vibration Hamiltonian.
Journal: submitted to J. Mol. Spectrocs.
YEAR: 1999
(MASIE)Subsection : 3.1 and 3.2
Abstract: Rotational structure of the triply degenerate vibrational state $\nu_6(F_{1u})$ of the octahedral molecule Mo(CO)$_6$ is analyzed qualitatively on the basis of classical mechanics. We show that the energy level redistribution between the vibrational components of $\nu_6(F_{1u})$ occurs due to rotational excitation and is related to the formation of singular points of classical rotational energy surfaces. The singularity is stable under small variations of parameters of the effective rovibrational Hamiltonian. Parameters responsible for the persistence of this phenomenon are specified. Comparison with quantum calculations demonstrates the high qualitative and quantitative accuracy of our classical analysis.

Author(s): H. R. Dullin and R. W. Easton.
Title: Stability of Levitrons.
Journal: Physica D, 126, 1-17
YEAR: 1999
(MASIE)Subsection: 1.1
Abstract: The Levitron is a magnetic spinning top which can levitate in the constant field of a repelling base magnet. An explanation for the stability of the Levitron using an adiabatic approximation has been given by M.V. Berry. In experiments the top eventually loses stability at a critical spin rate which can not be predicted by Berry's approach. The present work develops an exact theory of the Levitron with six degrees of freedom which allows for the calculation of the critical spin rate. The main result is a complete classification of possible Levitrons that allow for an interval of stable spin rates. Stability of the relative equilibrium is lost in Hamiltonian Hopf bifurcations if either the spin rate is too large or too small.

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Author(s): J. Izaguirre, S. Reich, and R.D. Skeel
Title: Longer time steps for molecular dynamics
Journal: J. Chem. Phys. 110 , 9853-9864, 1999.
YEAR: 1999
(MASIE)Subsection : 2.2

Authors:   J. S. W. Lamb and I. Melbourne
Title:     Bifurcation from discrete rotating waves
Journal:   Arch. Rat. Mech. Anal., 149, 229--270
YEAR: 1999
(MASIE)Subsection : 1.1
Abstract: Discrete rotating waves are periodic solutions that have discrete spatiotemporal symmetries in addition to their purely spatial symmetries.We present a systematic approach to the study of local bifurcation from discrete rotating waves.   The approach centers around the analysis of diffeomorphisms that are equivariant with respect to distinct group actions in the domain and the range.
Our results are valid for dynamical systems with finite symmetry group, and more generally for bifurcations from isolated discrete rotating waves in dynamical systems with compact symmetry group.
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Author(s): J.S.W. Lamb and R.M. Roberts.
Title: Reversible equivariant linear systems.
Journal: J. Diff. Eq. 152, 239-279.
YEAR: 1999
(MASIE)Subsection : Related
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Author(s): H. Hanßmann
Title: Quasi-periodic Motions of a Rigid Body II--- Implications for the Original System
Preprint: RWTH Aachen
YEAR: 1999
(MASIE)Subsection : Related publication
Abstract: This is a sequel to
Quasi-periodic Motions of a Rigid Body I--- Quadratic Hamiltonians on the Sphere with a Distinguished Parameter.
The original system, while being an $\varepsilon$-perturbation of the Euler top, is $\varepsilon^2$-close to its normal form approximation. The normal form automatically `removes the degeneracy' of the superintegrable Euler top and KAM-theory allows to conclude that a large part of the phase space is filled by Cantor families of invariant 3-tori. The way these 3-tori are distributed in phase space is determined by persisting invariant 2-tori, serving as `landmarks' in the same way as the equilibria did for the one-degree-of-freedom systems treated in Quasi-periodic Motions of a Rigid Body I.
The rigid body motion along such 2-tori closely follows the rotational-precessional motion of the Euler top.
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Author(s): Y. Houndonougbo, B. Laird, and B. Leimkuhler
Title: Molecular dynamics algorithms for mixed hard-core/continuous potentials
Journal:J. Mol. Phys. (to appear)
YEAR: 1999
(MASIE)Subsection : 2.2

Author(s): I.N. Kozin, R.M. Roberts and J. Tennyson
Title: Symmetry and structure of H$_3^+$
Journal: J. Chem. Phys., 111, 140-150 (1999).
YEAR: 1999
(MASIE)Subsection : 3.2
Abstract: We present a global study of how the relative equilibria of the H$_3^+$ ion change as the angular momentum $J$ increases. A relative equilibrium is a classical trajectory for which the molecule rotates about a stationary axis without changing its shape. The study confirms previous results which show that the geometry of the minimum energy relative equilibria changes from an equilateral triangle toa symmetric linear configuration at around $J$ = 47. The series of bifurcations and stability changes that accompany this transition is presented in detail. New results include the discovery that the rotating equilateral triangle remains linearly stable for a large range of angular momentum values beyond the point where it ceases to be a minimum of the total energy. A third type of relative equilibrium, a
rotating isosceles triangle, is also found to be linearly stable in the approximate range $J$ = 0---34. Both the equilateral and isosceles triangle configurations lose stability via Hamiltonian-Hopf bifurcation. The frequencies and symmetry species of the normal modes of the stable relative equilibria are computed and harmonic quantization is used to predict how the symmetries of the lowest lying quantum states will change as $J$ increases. Energy level clustering due to tunnelling between symmetry-equivalent relative equilibria is described.

Author(s): I.N. Kozin, R.M. Roberts and J. Tennyson
Title: Relative equlibria of D$_2$H$^+$ and H$_2$D$^+$
Journal: Mol. Phys. (in press)
YEAR: 1999
(MASIE)Subsection : 3.2
Abstract: Relative equilibria of molecules are classical trajectories corresponding to steady rotations about stationary axes during which the shape of the molecule does not change. They can be used to explain and predict features of quantum spectra at high values of the total angular momentum $J$ in much the same way that absolute equilibria are used at low $J$. This paper gives a classification of the symmetry types of relative equilibria of AB$_2$ molecules and computes the relative equilibria bifurcations diagrams and molecules and computes the relative equilibria bifurcations diagrams and
normal mode frequencies for \D$_2$H$^+$\ and H$_2$D$^+$. These are then fed into a harmonic quantization procedure to produce a number of predictions concerning the structures of energy level clusters and their rearrangements as $J$ increases. In particular the formation of doublet pairs is predicted for H$_2$D$^+$\ from $J \approx 20$.

Author(s): Leimkuhler, B.
Title: Reversible adaptive regularization methods for atomic N-body problems in applied fields
Journal: Appl. Num. Math. 29, 31-43
YEAR: 1999
(MASIE)Subsection : 2.2

Author(s): Leimkuhler, B.
Title: Reversible adaptive regularization: perturbed Kepler motion and classical atomic trajectories
Journal: Phil Trans. Roy. Soc.
YEAR: 1999
(MASIE)Subsection : 2.2

Author(s): G.Iooss
Title: Travelling waves in the Fermi-Pasta-Ulam lattice.
Preprint: INLN 99.25
YEAR: 1999
(MASIE)Subsection: Related publication
Abstract: We consider travelling wave solutions on a one-dimensional lattice, corresponding to mass particles interacting nonlinearly with their nearest neighbor (Fermi-Pasta-Ulam model). A constructive method is given, for obtaining all small bounded travelling waves for generic potentials, near the first critical value of the velocity. They all are solutions of a finite dimensional reversible ODE. In particular, near (above) the first critical velocity of the waves, we construct the solitary waveswhose global existence was proved by Friesecke et Wattis [1], using a variational approach. In addition, we find other travelling waves like (i) superposition of a periodic oscillation with a non zero averaged stretching or compression between particules, (ii) mainly localized waves which tend to uniformly stretched or compressed lattice at infinity, (iii) heteroclinic solutionsconnecting a stretched pattern with a compressed one.
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Author(s): J. Montaldi and R.M. Roberts
Title: Relative equilibria of molecules
Journal: J. Nonlinear Science 9, 1999, 53-88.
YEAR: 1999
(MASIE)Subsection : 1.1 and 3.2
Abstract: We apply and extend results from the paper "Persistence and Stability of Relative Equilibria" (above) to prove the existence of relative equilibria of molecules for small angular momentum. In this case, relative equilibria are pure rotational motions. For a generic molecule, there are 6 rotational modes, as for the rigid body. If the molecule is symmetric (e.g. methane), we show that there are (many) more, classifying them by their symmetry. We also consider their stability. For example, for methane, which has tetrahedral symmetry, there are 26 rotational modes, of which 6 are Lyapounov stable, 8 are linearly stable and 12 are unstable, corresponding to the three distinct symmetry types.
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Author(s): J. Montaldi
Title: Perturbing a symmetric resonance: the magnetic spherical pendulum
In: SPT98 -- Symmetry and Perturbation Theory II. A. Degasperis and G. Gaetaeds., World Scientific, to appear (1999).
YEAR: 1999
(MASIE)Subsection : Related publication
Abstract: The periodic orbits of the spherical pendulum are well-known. There are two types of modes: the planar oscillations and the circular modes (discovered by Huygens). The stable equilibrium is resonant, in that it has a pair of double eigenvalues. Adding a (constant vertical) magnetic field breaks the reflexional symmetry and so breaks the resonance. Indeed, one of the Huygens modes will rotate faster than the other. Moreover the planar modes will not persist as such, although the corresponding family of periodic orbits will continue to exist, albeit slightly perturbed. The object of this note is to describe the fate of the families of periodic orbits as the magnetic field is increased from 0. We assume that the magnetic perturbation preserves the rotational symmetry. The method is also applied to the analogous symmetry-breaking problem with square symmetry rather than the full rotation group. Singularity theoretic and normal form techniques are used.
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Author(s): S. Reich
Title: Preservation of adiabatic invariants under symplectic discretization
Journal: Applied Numerical Mathematics 29, 45-56, 1999.
YEAR: 1999
(MASIE)Subsection : 2.1

Author(s): S. Reich
Title: Backward error analysis for numerical integrators
Journal: SIAM J. Numer. Anal. 36 , 1549-1570, 1999
YEAR: 1999
(MASIE)Subsection : 2.1

Author(s): S. Reich
Title: Multiple time-scales in classical and quantum-classical molecular dynamics
Journal: Comput. Phys. 151 , 49-73, 1999.
YEAR: 1999
(MASIE)Subsection : 2.2

Author(s): S. Reich
Title: Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations.
Journal: J. Comput. Phys., to appear.
YEAR: 1999
(MASIE)Subsection : 2.3

Author(s): S. Reich
Title: Finite volume methods for multi-symplectic PDEs
Journal: BIT (to appear)
YEAR: 1999
(MASIE)Subsection : 2.3

Author(s): Roberts M. and Sousa Dias, M.E..
Title: Symmetries of Riemann ellipsoids.
Journal: Resenhas- IME. USP, Vol. 4, 2, pp 183-221
YEAR: 1999
(MASIE)Subsection : 4.4
Abstract: The results of Dirichlet, Dedekind and Riemann on " ellipsoidal figures of equilibrium'" of rotating self-gravitating fluids are reviewed in the context of the geometric theory of Hamiltonian systems with symmetry. In particular Riemann's classification is derived using only the existence of physically natural rotational symmetries, and so is shown to be applicable to models of liquid drops, atomic nuclei and elastic bodies as well as self-gravitating fluids. Similarly Dedekind's transposition symmetry is obtained as a simple consequence of the rotational symmetries. A detailed description is given of a generalization of \lq self-adjoint' ellipsoids. The symmetry groups of the different types of ellipsoidal figures of equilibrium are also computed.

Author(s): D.A.Sadovskii, B. Zhilinskii.
Title: Monodromy, diabolic points, and angular momentum coupling.
Journal: Phys. Lett. A., 256, 235 - 244
YEAR: 1999
(MASIE)Subsection : 1.2 , 3.1 and 3.3
Abstract: Monodromy, or the most basic obstruction to global action-angle coordinates is shown to be present in the well known problem of two coupled angular momenta. It is also shown that in the corresponding quantum problem monodromy manifests itself as the redistribution of energy levels between different multiplets of the quantum spectrum.
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Author(s): Sousa Dias, M.E..
Title: Relative equilibria in elasticity.
Journal: STP98- Symmetry and Pertubation Theory II, A. Degasperis and G. Gaeta eds., World Scientific, 258-267
YEAR: 1999
(MASIE)Subsection : 4.3 and 4.4
Abstract: Relative equilibria for Hamiltonian dynamical systems modelling the motion of hyperelastic, homogeneous, frame indifferent and isotropic bodies, are studied. We find conditions on the potential energy function $V$ for the existence of relative equilibria with certain prescribed symmetries, namely for those of type $S$.
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Author(s): Ch. Van Hecke, D.A.Sadovskii, B. Zhilinskii.
Title: Qualitative analysis of molecular rotation starting from inter-nuclear potential.
Journal: Europ. Phys. J. D, 7 ,199-209
YEAR: 1999
(MASIE)Subsection : 3.1 and 3.2
Abstract: We study how qualitative features of the molecular dynamics can be derived directly from the internuclear (vibrational) potential. This approach is presented on the example of a tetrahedral molecule A$_4$ using several increasingly elaborated models of the potential.
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Author(s): VU NGOC, San
Title: Quantum monodromy in integrable systems.
Journal: Communications in Mathematical Physics 203(2) pp.465--479
YEAR: 1999
(MASIE)Subsection : 3.1 and 3.3
Abstract: Let $P_1(h),\dots,P_n(h)$ be a set of commuting self-adjoint $h$-pseudodifferential operators on an $n$-dimensional manifold. If the joint principal symbol $p$ is proper, it is known from the work of Colin de Verdière and Charbonnel that in a neighbourhood of any regular value of $p$, the joint spectrum locally has the structure of an affine integral lattice. This leads to the construction of a natural invariant of the spectrum, called the quantum monodromy. We present this construction here, and show that this invariant is given by the classical monodromy of the underlying Liouville integrable system, as introduced by Duistermaat. The most striking application of this result is that all two degree of freedom quantum integrable systems with a focus-focus singularity have the same non-trivial quantum monodromy. For instance, this proves a conjecture of Cushman and Duistermaat concerning the quantum spherical pendulum.

Author(s): Zhilinskii B. I., Kozin I., and Petrov S.
Title: Correlation between asymmetric and spherical top: Imperfect quantum bifurcations.
Journal: Spectrochim. Acta A, 55, No 7-8, 1471-1484
YEAR: 1999
(MASIE)Subsection : 3.2
Abstract: The rotational energy level structures of quazi-spherical top molecules is investigated through the analysis of systems of stationary points on classical rotational energy surface. The series of simplest typical bifurcations of stationary points are given as a quasi-spherical molecule evolves to the spherical top limit due to rotational excitation or molecular isotopomerization. In this way the correlation between asymmetric and spherical top rotation energy multiplets is studied for A$_4$ and AB$_4$ molecules and corresponding isotopomers. It is demonstrated that the correlation depends on the point symmetry of the asymmetric top molecule. Slight symmetry breaking from C$_{2v}$ point symmetry down to C$_s$ results in the appearance of imperfect bifurcations The effect of imperfect bifurcation and its manifestation in molecular rotational spectra are discussed.

Author(s): J. Zhou, S. Reich and B.R Brooks
Title: Elastic molecular dynamics with self-consistent flexible constraints
Journal: J. Chem. Phys., to appear.
YEAR: 1999
(MASIE)Subsection : 2.2

1998

Author(s):    G. Benettin, F. Fassò and M. Guzzo
Title: Nekhoroshev-stability of L4 and L5 in the spatial restricted three-body problem
Journal: Regular and Chaotic Dynamics 3, 56-72 (1998)
YEAR: 1998
(MASIE)Subsection: 1.1
Abstract: We show that $L_4$ and $L_5$ in the spatial restricted circular three--body problem are Nekhoroshev--stable for all but a few values of the reduced mass up to the Routh critical value. This result is based on two extensions of previous results on Nekhoroshev--stability of elliptic equilibria, namely to the case of directional quasi-convexity, a notion introduced here, and to a (non-convex) steep case. We verify that the hypotheses are satisfied for $L_4$ and $L_5$ by means of numerically constructed Birkhoff
normal forms.
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Author(s): S. Bond and B. Leimkuhler.
Title: Time transformations for reversible variable stepsize integration.
Journal: Numerical Algorithms 19, 55-71, 1998.
YEAR: 1998
(MASIE)Subsection: 2.2

Author(s): Brack, M., Stephen C. Creagh and J. Law
Title: Spectral Fluctuations at the Bottom of a Potential
Journal: Phys. Rev. A, 57, 788 (1998)
(MASIE)Subsection: 3.3
Abstract: We evaluate trace formulas for various perturbations of two-dimensional harmonic oscillators. Such systems arise naturally in the expansion of generic potentials about local minima. For large enough perturbations, the usual theory for isolated orbits applies and we can reproduce the long and medium-range
oscillations in the density of states in terms of the shortest periodic orbits. For small perturbations, or low energies, the Gutzwiller amplitudes diverge due to the approaching degeneracy of the harmonic oscillator. We employ a perturbative analysis of the classical dynamics to give a treatment of the trace formula that is valid near the degenerate harmonic regime. First order perturbation theory works for generic cases. For certain potentials, such as Hénon-Heiles, discrete symmetries lead to a null result at first order and second order calculations are necessary to capture the dominant features.

Author(s): H. Broer, I. Hoveijn and M. van Noort
Title: A reversible bifurcation analysis of the inverted pendulum
Journal: Physica D, 112, 50--63.
YEAR: 1998
(MASIE)Subsection: 1.1

Author(s): H. Broer, G.A. Lunter and G.Vegter
Title: Equivariant singularity theory with distinguished parameters, two case studies of resonant Hamiltonian systems
Journal: Physica D, 112, 64 - 80.
YEAR: 1998
(MASIE)Subsection: 1.2

Author(s): H. Broer, I. Hoveijn, G.A. Lunter and G.Vegter
Title: Resonances in a Spring - Pendulum: algorithms for equivariant singularity theory
Journal: Nonlinearity, 11 (5), 1-37.
YEAR: 1998
(MASIE)Subsection: Related publ.

Author(s): H. Broer, C. Simó
Title: Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena
Journal: Bol. Soc. Bras. Mat, 29, 253--293.
YEAR: 1998
(MASIE)Subsection: Related publ.

Author(s): M S Child
Title: Quantum states in a champagne bottle
Journal: J Phys A, 31, 657
YEAR: 1998
(MASIE)Subsection : 2.2

Author(s): Creagh, S.
Title: Tunnelling in Two Dimensions
Book:   in S. Tomsovic and A. Legget eds. , Tunnelling in Complex Systems, World Scientific 1998.
YEAR: 1998
(MASIE)Subsection : 3.3
Abstract: An exploration is made of the behaviour of tunnelling phenomena in low-dimensional systems taking into account classical limits with varying degrees of nonintegrability, spanning the spectrum from completely integrable to completely chaotic motion. There are three broad categories: (a) integrable and KAM systems, for which tunnelling is intimately related to the complexification of the invariant tori that characterise classical mechanics; (b) mixed systems in which coupling to a chaotic sea is important and tunnelling rates are irregular and (c) completely chaotic problems in which tunnelling rate fluctuations can be analysed in terms of complex classical orbits. Between each of these there is a certain amount of overlap in the methods used and in the structures that are important, but there are also some very stark differences. Such connections are emphasised in the discussion.

Author: R. Cushman
Title: Routh's sphere
Journal: Reports on Mathematical Physics , 42, 47-70
YEAR: 1998
(MASIE)Subsection : 1.5
Abstract: In this paper we show that the integral map of Routh's sphere has monodromy when the sphere becomes gyroscopically unstable. This uses the non-Hamiltonian monodromy theorem of Cushman and Duistermaat.
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Author(s): R. Cushman, D. Kemppainen and J. Sniatycki
Title: A classical particle with spin realized by reduction of a nonlinear nonholonomic constraint.
Journal: Reports on Mathematical Physics, 41, 133-142
YEAR: 1998
(MASIE)Subsection: 1.5
Abstract: This paper describes the motion of a nonlinear nonholonomically constrained system which after reduction realizes a nonrelativistic classical particle with spin.
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Author(s): P. Duarte, R. L. Fernandes and W.M. Oliva
Title: Hamiltonian dynamics of the Lotka -Volterra Equations.
Journal: Jounal of Differential equations, 149, 143-189,1998
YEAR: 1998
(MASIE)Subsection : 1.4
Abstract: We show that for stable dissipative Lotka-Volterra systems the dynamics on the attractor are hamiltonian and we argue that complex dynamics can occur.

Author(s):   F. Fassò, M. Guzzo and G. Benettin
Title: Nekhoroshev--stability of elliptic equilibria of Hamiltonian systems
Journal: Communications in Mathematical Physics 197, 347-360 (1998)
YEAR: 1998
(MASIE)Subsection: 1.1
Abstract: We prove a conjecture by N.N. Nekhoroshev about the long--time stability of elliptic equilibria of Hamiltonian systems,
without any Diophantine condition on the frequencies. Higher order terms of the Hamiltonian are used to provide convexity. The
singularity of the action-angle coordinates at the origin is overcome by working in cartesian coordinates.
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Author(s):   F. Fassò and and T. Ratiu
Title: Compatibility of symplectic structures adapted to noncommutatively integrable systems
Journal: Journal of Geometry and Physics 27, 199-220 (1998).
YEAR: 1998
(MASIE)Subsection: Related publication
Abstract: It is known that any integrable, possibly degenerate, Hamiltonian system is Hamiltonian relative to many different symplectic structures; under certain hypotheses, the `semi--local' structure of these symplectic forms, written in local coordinates of action--angle type, is also known. The purpose of this paper is to characterize from the point of view of symplectic geometry the family of all these structures. The approach is based on the geometry of noncommutatively integrable systems and extends a recent treatment of the nondegenerate case by Bogoyavlenskij. Degenerate systems are comparatively richer in symplectic structures than nondegenerate ones and this has the counterpart that the bi--Hamiltonian property alone does not imply integrability. However, integrability is still guaranteed if a system is Hamiltonian with respect to three suitable symplectic structures. Moreover, some of the properties of recursion operators are retained.
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Author(s):   F. Fassò
Title: Quasi--periodicity of motions and complete integrability of Hamiltonian systems
Journal: Ergodic Theory and Dynamical Systems 18, 1349-1362 (1998)
YEAR: 1998
(MASIE)Subsection: Related publication
Abstract: Consider a Hamiltonian system with $d$ degrees of freedom whose motions are all linear on tori of some fixed dimension $n\le d$; is such a system necessarily completely (or else non--commutatively) integrable? We show that the answer is affermative under quite broad conditions, but not always, and we provide counterexamples.
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Author(s): R. L. Fernandes and W.M. Oliva
Title: Hamiltonian dynamics of the Lotka -Volterra Equations.
In book: International Conference on Differential Equations (Eqadiff 95),Eds. L. Magalhaes, C. Rocha, L.Sanchez, World Scientific, 327--334.
YEAR: 1998
(MASIE)Subsection : 1.4
Abstract: In this paper we discuss several aspects of the Hamiltonian structure of the Lotka- Volterra equations. In particular we show that the dynamics on the attractor are hamiltonian.

Author(s): G. Fusco and W.M. Oliva
Title: Asymptotic behaviour of a system of repelling particles: Asymptotic velocities, phases, scattering operators and integrability.
In book: International Conference on Differential Equations (Eqadiff 95),Eds. L. Magalhaes, C. Rocha, L.Sanchez, World Scientific, 69-81.
YEAR: 1998
(MASIE)Subsection : related Publication
Abstract: Asymptotic properties of the dynamics of a system of repelling particles are related to the asymptotic smoothness of the corresponding potential. In other words, the asymptotic velocities and phases, the scattering operator and the Liouville integrability of a system of repelling particles under the action of a (bounded below) potential are related to the range of the potential.

Author(s):    M. Guzzo, F. Fassò and G. Benettin
Title: On the stability of elliptic equilibria
Journal: MPEJ - Mathemathical Physics Electronic Journal 4, Paper 1 (1998).
YEAR: 1998
(MASIE)Subsection: 1.1
Abstract: We consider stability of elliptic equilibria in Hamiltonian systems in the frame of Nekhoroshev's theory, recovering the steepness assumption, in the form of convexity, from an appropriate treatment of the higher orders. The singularity of the action--angle coordinates is overcome by using Cartesian coordinates. We introduce an essential refinement of the perturbative technique used in a previous work on the subject, and obtain significant improvements of results, namely better values of the exponents controlling the stability time and the confinement around equilibrium, in case the equilibrium frequencies satisfy stronger nonresonance conditions. Within the same nonresonance assumptions the new method provides instead independent informations, namely one gets a better confinement on a reduced time scale.
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Author(s): H. Hanßmann
Title: The Reversible Umbilic Bifurcation
Jounal: Physica D, 112, 81-94
YEAR: 1998
(MASIE)Subsection : 1.1
Abstract: Hamiltonian systems with several degrees of freedom regularly lead to the investigation of bifurcating equilibria in reduced one-degree-of-freedom systems. This paper concerns equilibria with vanishing linearization, a co-dimension
two phenomenon in the reversible context. Under appropriate transversality conditions such equilibria have versal
unfoldings related to the elliptic and hyperbolic umbilic catastrophes. This has applications to gyrostat motion and also helps to explain the dynamics defined by the normal form of the Hénon-Heiles system. The occurring unfoldings turn out to be versal even in the general reversible context of not necessarily Hamiltonian systems.

Author(s): H. Hanßmann
Title: The Quasi-Periodic Centre-Saddle Bifurcation
Jounal: Journal of Differential Equations, 142, 305-370
YEAR: 1998
(MASIE)Subsection : 1.1 and 1.2
Abstract: Nearly integrable families of Hamiltonian systems are considered in the neighbourhood of normally parabolic invariant tori. In the integrable case such tori bifurcate into normally elliptic and normally hyperbolic invariant tori. With a KAM-theoretic approach it is shown that both the normally parabolic tori and the bifurcation scenario survive a non-integrable
perturbation, parametrised by pertinent large Cantor sets. These results are applied to rigid body dynamics.

Author(s): Th. Holder, B. Leimkuhler, and S. Reich
Title: Explicit variable step-size and time-reversible integration
Journal: submitted
YEAR: 1998
(MASIE)Subsection : 2.2

Author(s): G.Iooss, K.Kirchgässner
Title: Travelling waves in a chain of coupled nonlinear oscillators.
Journal: C.R.Acad. Sci. Paris, 1998, 327, I , 855-860
YEAR: 1998
(MASIE)Subsection : Related publication
Abstract: In a chain of nonlinear oscillators, linearly coupled to their nearest neighbors, all travelling waves of small amplitude are found as solutions of finite dimensional reversible dynamical systems. The coupling constant and the inverse wave speed form the parameter space. The groundstate consists of a one-parameter family of periodic waves. It is realized in a certain parameter region containing all cases of light coupling. Beyond the border of this region the complexity of wave-forms increases via a succession of bifurcations. In this paper we give an appropriate formulation of this problem, prove the basic facts about the reduction to finite dimensions, show the existence of the ground states and discuss the first bifurcation by determining a normal form for the reduced system. Finally we show the existence of nanopterons,which are localized waves with a noncancelling periodic tail at infinity whose amplitude is exponentially small in the bifurcation parameter.
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Author(s): Leimkuhler, B.
Title: Symplectic methods for rigid body dynamics, Computational Molecular Dynamics: Challenges, Methods, Ideas.
Book: Lect. Notes in Computational Sci. and Engr., Springer, 1998.
YEAR: 1998
(MASIE)Subsection : 2.2

Author(s): B. Leimkuhler.
Title: Timestep Acceleration of Waveform Relaxation
Journal: SIAM J. on Numerical Analysis
YEAR: 1998
(MASIE)Subsection : 2.3

Author(s): H.Y. Mussa and J. Tennyson
Title: Calculation of rotation-vibration states of water at dissociation
Journal: J. Chem. Phys., 109, 10885-10892 (1998)
YEAR: 1998
(MASIE)Subsection : 3.1
Abstract: We present rotation-vibrational levels of water up to dissociation limit using two recent, global potential energy surfaces. These calculations are performed using our recently developed discrete variable representation (DVR) based parallel code (PDVR3D), which runs on computers with massively parallel processors. Variational tests on the convergence of these results show convergence within 0.5 cm$^{-1}$. Analysis of the highest wavefunctions for the vibrational
energy levels are also shown. Tests on previous calculations performed using conventional computers suggest that convergence for high-lying rotationally excited is not as good as claimed.

Author(s): P. Nettesheim and S. Reich
Title: Symplectic multiple-time-stepping integrators for quantum-classical molecular dynamics
Journal: Lecture Notes in Computational Science and Engineering, Vol. 4, 412-420, 1998.
YEAR: 1998
(MASIE)Subsection : 2.2

Author(s): R. Prosmiti, H.Y. Mussa and J. Tennyson
Title: Quantum states of triatomic molecules at dissociation
Book: Molecular Quantum States at Dissociation, R. Prosmiti, J. Tennyson and D.C. Clary (Eds.), (CCP6, Daresbury, 1998).
YEAR: 1998
(MASIE)Subsection : 3.3
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Author(s): Sadovskii D. A., and Zhilinskii B. I.
Title: Tuning the hydrogen atom in crossed fields between the Zeeman and Stark limits.
Journal: Phys. Rev. A, 57, 2867-2884
YEAR: 1998
(MASIE)Subsection : 3.3
Abstract: We consider the hydrogen atom in the orthogonal electric and magnetic fields whose strength is assumed to be small enough for the Coulomb $n$-shell perturbation to apply. Appropriate scaling of the two fields leads to a uniform parametrization of teh problem by $S$, the combined strength of teh two fields, and $\alpha$, the ratio of the two field strengths. The initial six-dimensional phase space $R_6$ is lifted to the standard Kustaanheimo-Stiefel eight-dimensional space and then reduced explicitly to the $S_2\times S_2$ reduced space of the $n$ shell using the Lie transformation to the third order in $S$. At fixed $S$ the system is uniformly tuned between the Zeeman and the Stark limits using the analytic formulas of the perturbation theory. The approach requires application of the invariant theory, group theory, and topology to the analysis of teh dynamics on the reduced space $S_2\times S_2$ and the subsequent explicit transition to the original $R_6$. In particular we follow the evolution of the four principal periodic orbits (nonlinear normal modes) and corresponding four equilibria on $S_2\times S_2$.
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Author(s): B. I. Zhilinskii, S.V. Petrov
Title: Complete bifurcation analysis of tetrahedral molecules A$_4$ and their isotopically substituted A(2)A(2)*.
Journal: Optika i Spektrosk. 85, No.3, (1998) (in Russian); English translation: Opt Spectrosc. 85, No.3, 360-364
YEAR: 1998
(MASIE)Subsection : 3.2

1997

Authors:   P. Ashwin and I. Melbourne
Title:     Noncompact drift for relative equilibria and relative periodic orbits
Journal:   Nonlinearity, 10, 595--616
YEAR: 1997
(MASIE)Subsection : 1.1
Abstract: : In the context of equivariant dynamical systems with a compact Lie group $\Gamma$ of symmetries, Field and Krupa have given sharp upper bounds on the drifts associated with relative equilibria and relative periodic orbits.   For relative equilibria consisting of points of trivial isotropy, the drifts correspond to tori in $\Gamma$. Generically, these are maximal tori. Analogous results hold when there is a nontrivial isotropy subgroup $\Sigma$, with $\Gamma$ replaced by $N(\Sigma)/\Sigma$.
In this paper, we generalize the results of Field and Krupa to noncompact Lie groups. The drifts now correspond to tori or lines (unbounded copies of $\R$) in $\Gamma$ and generically these are maximal tori or lines. Which of these drifts is preferred, compact or unbounded, depends on $\Gamma$: there are examples where compact drift is preferred (Euclidean group in the plane), where unbounded drift is preferred (Euclidean group in three dimensional space) and where neither is preferred (Lorentz group).
Our results partially explain the quasiperiodic (Winfree) and linear (Barkley) meandering of spirals in the plane, as well as the drifting
behavior of spiral bound pairs (Ermakova et al). In addition, we obtain predictions for the drifting of the scroll solutions (scroll waves and scroll rings, twisted and linked) considered by Winfree and Strogatz.
URL

Author(s): H.M.A. Castro and W.M. Oliva
Title: Poisson structures on the Phase Space of Mechanical Systems with Constraint
Journal: Resenhas IME-USP, Vol. 3, N. 1, 85--92.
YEAR: 1997
(MASIE)Subsection : 1.5
Abstract: The existence of a Poisson structure on the phase space of mechanical systems with a fixed constraint satisfying the geometrical property that any conservative mechanical system with this constraint is itself hamiltonian (with respect to the Poisson structure) implies the integrability of the constraint. Two others equivalent geometrical properties are also presented.

Author(s): Creagh, Stephen C. and Peter Dimon
Title: Geometrical Orbits in the Power Spectra of Waves
Journal: Phys. Rev. E , 55, 5551 (1997).
YEAR: 1997
(MASIE)Subsection : 3.3

Author(s): G. Fusco and W.M. Oliva
Title: Integrability of a System of N Electrons Subjected to Coulombian Interactions
Journal: Journal of Differential equations, vol.35, N. 1, 16--40.
YEAR: 1997
(MASIE)Subsection : Related Publication
Abstract: The Liouville integrability of a system of N repelling particles in $R^n$, for a large class of potentials, is obtained by showing that the asymptotic velocities are smooth first integrals, independent, and in involution. A new proof for the existence of the asymptotic velocities is also presented.

Authors:   M. Golubitsky, V. G. LeBlanc and I. Melbourne
Title:     Meandering of the spiral tip: an alternative approach
Journal:   J. Nonlin. Sci., 7, 557--586
YEAR : 1997
(MASIE)Subsection : 1.1
Abstract: Meandering of a one-armed spiral tip has been noted in chemical reactions and numerical simulations. Barkley, Kness and Tuckerman show that meandering can begin by Hopf bifurcation from a rigidly rotating spiral wave (a point that is verified in a B-Z reaction by Li, Ouyang, Petrov and Swinney). At the codimension two point where (in an appropriate sense) the frequency at Hopf bifurcation equals the frequency of the spiral wave, Barkley notes that spiral tip meandering can turn to linearly translating spiral tip motion.
Barkley also presents a model showing that the linear motion of the spiral tip is a resonance phenomenon, and this point is verified
experimentally by Li et al and proved rigorously by Wulff. In this paper we suggest an alternative development of Barkley's model
extending the center bundle constructions of Krupa from compact groups to noncompact groups and from finite dimensions to function spaces. This approach allows us to consider various bifurcations from a rotatingwave. In particular, we can analyze in a straightforward manner the codimension two Barkley bifurcation and the codimension two Takens-Bogdanov bifurcation from a rotating wave. We also discuss Hopf bifurcation from a many armed spiral showing that meandering and resonant linear motion of the spiral tip do not always occur.
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Author(s): H. Hanßmann
Title:Quasi-periodic Motions of a Rigid Body I
--- Quadratic Hamiltonians on the Sphere with a Distinguished Parameter
Journal: Regular and Chaotic Dynamics, 2(2), 41-57
YEAR: 1997
(MASIE)Subsection : Related Publication
Abstract: The motion of a dynamically symmetric rigid body, fixed at one point and subject to an affine (constant+linear) force field is studied. The force being weak, the system is treated as a perturbation of the Euler top, a superintegrable system. Averaging along the invariant 2-tori of the Euler top yields a normal form which can be reduced to one degree of freedom, parametrised by the corresponding actions. The behaviour of this family is used to identify quasi-periodic motions of the rigid body with two or three independent frequencies.

Author(s): S. Hansen, S.H{\o}rl{\"u}ck, D. Zauner, Dimon P., C. Ellegaard, and S.C.Creagh
Title: Geometrical Orbits of Surface Waves from a Circular Hydraulic Jump
Journal: Phys. Rev. E , 55, 7048 (1997).
YEAR: 1997
(MASIE)Subsection : 3.3

Author(s): A. Dullweber, B. Leimkuhler, and R. McLachlan
Title: Split-Hamiltonian methods for rigid-body molecular dynamics
Journal: J. Phys. Chem., 107:5840, 1997.
YEAR: 1997
(MASIE)Subsection : 2.2

Author(s): S. Hansen, S.H{\o}rl{\"u}ck, D. Zauner, Dimon P., C. Ellegaard, and S.C.Creagh
Title: Geometrical Orbits of Surface Waves from a Circular Hydraulic Jump
Journal: Phys. Rev. E , 55, 7048 (1997).
YEAR: 1997
(MASIE)Subsection : 3.3

Author(s): W. Haung, and B. Leimkuhler
Title: The Adaptive Verlet Method
Journal: SIAM Journal on Scientific Computing, 1997.
YEAR: 1997
(MASIE)Subsection : Related publ. (Section 2)

Author(s): A. Kol , B. Laird and B. Leimkuhler
Title: A symplectic method for rigid-body molecular simulation
Journal: J. Chem. Phys., 107:2580-2588, 1997.
YEAR: 1997
(MASIE)Subsection : 2.2

Author(s): B. Leimkuhler, J. Frank and W. Huang
Title: Geometric integrators for classical spin systems
Journal: Comput. Phys. 133, 160-172, 1997
YEAR: 1997
(MASIE)Subsection : 2.2

Author(s): B. Leimkuhler and E. Van Vleck
Title: Orthosymplectic Integration of Linear Hamiltonian Systems.
Journal: Numerische Mathematik, 1997
YEAR: 1997
(MASIE)Subsection : 2.3

Author(s): Meier, P., Matthias Brack and Stephen Creagh
Title: Semiclassical description of large multipole-deformed metal clusters
Book: Z.~Phys.~D , 41, 281 (1997)
YEAR: 1997
(MASIE)Subsection : 3.3

Author(s): Michel L. and Zhilinskii B. I.
Title: Rydberg states of atoms and molecules. Group theoretical and topological analysis.
Preprint: IHES/P/97/54.
YEAR: 1997
(MASIE)Subsection : 3.1 and 3.3
Abstract: Rydberg states of atoms and molecules are studied within the qualitative approach based primarily on topological and group theoretical analysis. The correspondence between classical and quantum mechanics is explored to apply the results of qualitative (topological) approach to classical mechanics developed by Poincar\'e, Lyapounov, Smale to quantum problems. The study of the action of the symmetry group of the problem considered on the classical phase space enables us to predict qualitative features of the energy level patterns for quantum Rydberg operators.

Author(s): J. Montaldi
Title: Persistence and stability of relative equilibria.
Journal: Nonlinearity 10, 1997, 449-466.
YEAR: 1997
(MASIE)Subsection : 1.1
Abstract: We consider relative equilibria in symmetric Hamiltonian systems, and their persistence or otherwise as the momentum is varied. The symmetry group in question is assumed to be compact. In particular, we extend a result about persistence of relative equilibria for values of the momentum map that are regular for the coadjoint action, to arbitrary values, provided that either the action on the phase space is locally free, or that the relative equilibrium is at a local extremum of the reduced Hamiltonian. We also consider the Lyapunov stability of such extremal relative equilibria.
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Author(s): J. Montaldi
Title: Persistance d'orbites périodiques relatives dans les systèmes hamiltoniens symétriques
Journal: Comptes Rendues de l'Acad. des Sciences 324 , 1997, 553-558.
YEAR: 1997
(MASIE)Subsection : 1.1
Abstract: It was known to Poincaré that a non-degenerate periodic orbit in a Hamiltonian system persists to nearby energy-levels. In this note, we consider the analogous problem for relative periodic orbits in symmetric Hamiltonian systems. We show that non-degenerate relative periodic orbits persist to nearby values of the energy-momentum map, under the hypothesis that the group of symmetries acts freely.
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Author(s): Roberts M. and Sousa Dias, M.E..
Title: Bifurcations from relative equilibria of Hamiltonian systems.
Journal: Nonlinearity 10, 1719-1738, 1997.
YEAR: 1997
(MASIE)Subsection : 1.1
Abstract: A symplectic version of the slice theorem for compact group actions is used to prove a bifurcation theorem for relative equilibria of symmetric Hamiltonian systems. The bifurcation theorem is applied to two examples, the classical dynamics of an $XY_2$ molecule near a symmetric linear equilibrium, and the dynamics of a system of two coupled identical axisymmetric rigid bodies near an equilibrium for which the two bodies are aligned on the top of each other. In addition reduction of these systems to appropriate "slices" is used to describe other aspects of their dynamics. The results suggest that this might be a useful general technique for describing the dynamics of systems near relative equilibria which are singular points of the associated momentum map.

Author(s): J.M. Rost, and G. Tanner
Title: Two electron atoms: from resonances to fragmentation
In Book: Classical, Semiclassical and Quantum Dynamics in Atoms, Eds.: H. Friedrich,
and B. Eckhardt, (Springer-Verlag, Berlin), p 274 -- 303
YEAR: 1997
(MASIE)Subsection : 3.3

Author(s): G. Tanner
Title: How chaotic is the stadium billiard? A semiclassical analysis
Journal: J. Phys. A. 30, p 2863 -- 2888
YEAR: 1997
(MASIE)Subsection : 3.3

Author(s): VU NGOC, San
Title: Formes normales semi-classiques des systèmes complètement intégrables au voisinage d'un point critique de l'application moment.
Preprint: Institut Fourier (to appear in Asympt. Analys.)
YEAR: 1997
(MASIE)Subsection : 1.2 and 3.3
Abstract: The semi-classical study of a 1-dimensional Schrödinger operator near a non-degenerate maximum of the potential has lead Colin de Verdière and Parisse to prove a microlocal normal form theorem for any 1-dimensional pseudo-differential operator with the same kind of singularity. We present here a generalization of this result to
pseudo-differential integrable systems of any finite degree of freedom with a Morse singularity. Our results are based upon Eliasson's classical mechanical study of critical integrable systems.