Publications
[2004] [2003][ 2002 ] [ 2001 ] [ 2000 ] [ 1999 ] [ 1998 ] [1997 ]
1. Geometry and Dynamics
1.1. Stability and bifurcations of REs and RPOs
2004
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.
YEAR: 2004
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
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.
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
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
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): Heinz Hanßmann and Jan-Cees van der Meer
Title: On Non-Degenerate Hamiltonian Hopf Bifurcations in 3DOF Systems
Preprint: RWTH Aachen
YEAR: 2003
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
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
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
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
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
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
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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2002
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
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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
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
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.
Author(s): H. Dullin and F. Fasso`
Title: An algorithm for detecting Directional Quasi-Convexity
Preprint:
YEAR: 2002
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
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): J. Lamb, C. Wulff
Title: Reversible relative periodic orbits
Journal: J. Diff. Eq. , 178, 60-100
YEAR: 2002
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): F. Laurent-Polz
Title: Point vortices on the sphere: a case with opposite vorticities
Journal: Nonlinearity, 15, 143--171.
YEAR: 2002
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): 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
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): G. Patrick, R.M. Roberts and C. Wulff
Title: Stability of Poisson equilibria and Hamiltonian relative equilibria by energy methods.
Preprint:
YEAR: 2002
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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
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
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): M. Rodríguez-Olmos and M. E. Sousa-Dias.
Title: A Note on the Existence and Symmetries of Relative Equilibria in Simple Mechanical Systems
Preprint: preprint
YEAR: 2002
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.
Author(s): Tanya Schmah
Title: A Cotangent Bundle Slice Theorem
Journal: (in preparation)
YEAR: 2002
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
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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2001
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
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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Author(s): H. Hanßmann
Title: A Survey on Bifurcations of Invariant Tori
Preprint: Princeton University
YEAR: 2001
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.
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
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
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.
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
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
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): Chjan Lim, James Montaldi and Mark Roberts
Title: Relative equilibria of point vortices on the sphere.
Jounal: Physica D 148, 97-135.
YEAR: 2001
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): 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
Author(s): Ortega, J.-P. and T.S. Ratiu
Title: The dynamics around stable and unstable Hamiltonian relative equilibria
Preprint:
YEAR: 2001
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
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Author(s): C. Wulff
Title: Persistence of relative equilibria in Hamiltonian systems with noncompact symmetries
Preprint: Freie Universitát Berlin, 2001
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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2000
Author(s): Pascal Chossat, Juan-Pablo Ortega and Tudor S. Ratiu
Title: Hamiltonian Hopf bifurcation with symmetry
Preprint:
YEAR: 2000
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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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
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): Juan-Pablo Ortega
Title: Relative normal modes for nonlinear Hamiltonian systems.
Preprint:
YEAR: 2000
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
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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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
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): Pascal Chossat, Debra Lewis, Juan-Pablo Ortega and Tudor S. Ratiu
Title: Bifurcation of relative equilibria in mechanical systems with symmetry
Preprint:
YEAR: 1999
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): H. R. Dullin and R. W. Easton.
Title: Stability of Levitrons.
Journal: Physica D, 126, 1-17
YEAR: 1999
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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Authors: J. S. W. Lamb and I. Melbourne
Title: Bifurcation from discrete rotating waves
Journal: Arch. Rat. Mech. Anal., 149, 229--270
YEAR: 1999
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. Montaldi and R.M. Roberts
Title: Relative equilibria of molecules
Journal: J. Nonlinear Science 9, 1999, 53-88.
YEAR: 1999
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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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
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): 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. Hanßmann
Title: The Reversible Umbilic Bifurcation
Jounal: Physica D, 112, 81-94
YEAR: 1998
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: Jounal 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): 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
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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
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): 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
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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1997
Authors: P. Ashwin and I. Melbourne
Title: Noncompact drift for relative equilibria and relative periodic orbits
Journal: Nonlinearity, 10, 595--616
YEAR: 1997
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.
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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
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): J. Montaldi
Title: Persistence and stability of relative equilibria.
Journal: Nonlinearity 10, 1997, 449-466.
YEAR: 1997
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
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
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.