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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: A connection approach to Lie systems
In: Publicaciones de la RSME 6 (2004), 235--239.
YEAR: 2004
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.
2003
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
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
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: 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
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.
2002
Author(s): Birtea, P., M. Puta, and T.S. Ratiu
Title: Controllability of reduced systems
Preprint:
YEAR: 2002
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
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
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
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): 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
Author(s): W.M. Oliva
Title: Morse-Smale semiflows. Openess and A-stability
Journal: Fields Inst. Comm. 31, AMS, 285-307
YEAR: 2002
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
Authors: F.R. Austin & T.J. Bridges
Title: A bundle view of boundary value problems: generalizing the Gardner-Jones bundle.
Journal: Preprint
Year: 2001
Authors: T.J. Bridges & S. Reich
Title: Computing Lyapunov exponents on a Stiefel manifold
Journal: PhysicaD 156: 219-238
Year: 2001
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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): P.-L. Buono, J.S.W. Lamb & R.M. Roberts.
Title: Steady-state bifurcations in reversible equivariant systems.
Preprint:
YEAR: 2001
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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
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
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
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): 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
Author(s): L. Fichmann and W.M. Oliva
Title: One-to-oneness and hyperbolicity
Journal: Fields Inst. Comm. 29, AMS, 113-131
YEAR: 2001
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
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.
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.
2000
Author(s): H.M.A. Castro, M:H. Kobayashi and W.M. Oliva
Title: Partially hyperbolic $\Sigma$-geodesic flows.
Journal: Journal of Differential equations, (to appear).
YEAR: 2000 (to appear)
Abstract: The paper deals with examples of partially flows motivated by the study of the $\Sigma$- geodesic flows corresponding to (second order) mechanical system with constraint. Suitable conditions properly decoup its variational equation and imply the hyperbolic properties of the trajectories. The case of a general Lie group and of a semi-simple Lie group are analyzed.
Author(s): I. Kupka and W.M. Oliva
Title: The Non-Holonomics Mechanics.
Journal: Jounal of Differential Equations, (to appear).
YEAR: 2000
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.
Author(s): James Montaldi and Mark Roberts
Title: A note on semisymplectic actions of Lie groups
Jounal: C. R. Acad. Sci. Paris 330, 1079-1084.
YEAR: 2000
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.
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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
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.
1999
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
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): 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): J.S.W. Lamb and R.M. Roberts.
Title: Reversible equivariant linear systems.
Journal: J. Diff. Eq. 152, 239-279.
YEAR: 1999
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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
Keywords: computation, Hamiltonian systems, reversible dynamics
Author(s): H. Hanßmann
Title: Quasi-periodic Motions of a Rigid Body II--- Implications for the Original System
Preprint: RWTH Aachen
YEAR: 1999
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): G.Iooss
Title: Travelling waves in the Fermi-Pasta-Ulam lattice.
Preprint: INLN 99.25
YEAR: 1999
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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1998
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
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
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
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
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): 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): 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
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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1997
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
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.
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
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): W. Haung, and B. Leimkuhler
Title: The Adaptive Verlet Method
Journal: SIAM Journal on Scientific Computing, 1997.
YEAR: 1997
Keywords : computation, Hamiltonian systems, reversible dynamics, N-body problems