Continuation of periodic orbits in conservative and Hamiltonian systems

Muñoz-Almaraz, F. J.; Freire, Emilio; Galán, J.; Doedel, Eusebius J.; Vanderbauwhede, André

doi:10.1016/s0167-2789(03)00097-6

Cited by 120 publications

(118 citation statements)

References 18 publications

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“…Remark 3.13 Proposition 3.12 c) ii) strengthens a related result of Muñoz-Almaraz et al [29,Proposition 12]. In our notation, they show that the algebraic multiplicity m a of the eigenvalue 1 of the linearization DΦT (x) at aT -periodic orbit of a Γ-symmetric Hamiltonian system satisfies m a ≥ (dim Γ + 1) + (dim Z + 1) where Z is the centre of Γ.…”

Section: Linearization Along Non-degenerate Rpossupporting

confidence: 80%

“…Moreover we have the following convergence theorem which generalizes corresponding results in [15,29,35] to periodic orbits with spatio-temporal symmetry:…”

Section: Remark 24mentioning

confidence: 55%

“…Existing powerful continuation methods for periodic orbits, see e.g. [15,29,38] and references therein, are readily applicable in our approach.…”

Section: Remark 24mentioning

confidence: 99%

“…[15,29]. In this paper we are concerned with the numerical continuation of symmetric Hamiltonian periodic orbits and RPOs.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, in the case of continuous symmetries, we compute symmetry breaking bifurcations of relative periodic orbits from periodic orbits and show how to continue non-degenerate RPOs of compact group actions in the conserved quantities momentum and energy, building on the persistence results from [34]. We use the methods of unfolding parameters of Muñoz-Almaraz et al [15,29], but whereas Muñoz-Almaraz et al [15,29] continue Hamiltonian periodic orbits in external parameters we focus on continuation in internal parameters, namely the conserved quantities of the Hamiltonian system. The main issue here is to specify how unfolding parameters have to be added in order to compute the whole manifold of nearby RPOs without computing symmetry-conjugate solutions.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Continuation of Hamiltonian Relative Periodic Orbits

Wulff

Schebesch

2008

J Nonlinear Sci

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The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years there has been rapid progress in the development of a bifurcation theory for dynamical systems with structure, such as symmetry or symplecticity.But as yet there are few results on the numerical computation of those bifurcations. The methods we present in this paper are a first step towards a systematic numerical analysis of generic bifurcations of Hamiltonian symmetric periodic orbits and relative periodic orbits (RPOs). First we show how to numerically exploit spatio-temporal symmetries of Hamiltonian periodic orbits. Then we describe a general method for the numerical computation of RPOs persisting from periodic orbits in a symmetry breaking bifurcation. Finally we present an algorithm for the numerical continuation of non-degenerate Hamiltonian relative periodic orbits with regular drift-momentum pair. Our pathfollowing algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincaré section and a tangential continuation method with implicit reparametrization. We apply our methods to continue the famous Figure Eight choreography of the three-body system. We find a relative period doubling bifurcation of the planar rotating Eight family and compute the rotating choreographies bifurcating from it.

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Section: Linearization Along Non-degenerate Rpossupporting

confidence: 80%

“…Moreover we have the following convergence theorem which generalizes corresponding results in [15,29,35] to periodic orbits with spatio-temporal symmetry:…”

Section: Remark 24mentioning

confidence: 55%

“…Existing powerful continuation methods for periodic orbits, see e.g. [15,29,38] and references therein, are readily applicable in our approach.…”

Section: Remark 24mentioning

confidence: 99%

“…[15,29]. In this paper we are concerned with the numerical continuation of symmetric Hamiltonian periodic orbits and RPOs.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical Continuation of Hamiltonian Relative Periodic Orbits

Wulff

Schebesch

2008

J Nonlinear Sci

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Periodic Orbits of Hamiltonian Systems

Sbano

2009

Encyclopedia of Complexity and Systems Science

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Numerical Continuation Methods for Dynamical Systems

Krauskopf¹,

Osinga²,

Galán-Vioque³

2007

Understanding Complex Systems

205

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Springer Complexity is an interdisciplinary program publishing the best research and academic-level teaching on both fundamental and applied aspects of complex systems -cutting across all traditional disciplines of the natural and life sciences, engineering, economics, medicine, neuroscience, social and computer science.Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations of which are the spontaneous formation of distinctive temporal, spatial or functional structures. Models of such systems can be successfully mapped onto quite diverse "real-life" situations like the climate, the coherent emission of light from lasers, chemical reaction-diffusion systems, biological cellular networks, the dynamics of stock markets and of the internet, earthquake statistics and prediction, freeway traffic, the human brain, or the formation of opinions in social systems, to name just some of the popular applications.Although their scope and methodologies overlap somewhat, one can distinguish the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence.The two major book publication platforms of the Springer Complexity program are the monograph series "Understanding Complex Systems" focusing on the various applications of complexity, and the "Springer Series in Synergetics", which is devoted to the quantitative theoretical and methodological foundations. In addition to the books in these two core series, the program also incorporates individual titles ranging from textbooks to major reference works.

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Continuation of periodic orbits in conservative and Hamiltonian systems

Cited by 120 publications

References 18 publications

Numerical Continuation of Hamiltonian Relative Periodic Orbits

Numerical Continuation of Hamiltonian Relative Periodic Orbits

Periodic Orbits of Hamiltonian Systems

Numerical Continuation Methods for Dynamical Systems

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