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Periodic Orbits of Hamiltonian Systems

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Mathematics of Complexity and Dynamical Systems

Article Outline

Glossary

Definition

Introduction

Periodic Solutions

Poincaré Map and Floquet Operator

Hamiltonian Systems with Symmetries

The Variational Principles and Periodic Orbits

Further Directions

Acknowledgments

Bibliography

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Abbreviations

Hamiltonian:

are called all those dynamical systems whose equations of motion form a vector field X H defined on a symplectic manifold (\({\mathcal{P},\omega}\)), and X H is given by \({i_{X_H}\omega=\text{d} H}\), where \({H\colon\mathcal{P}\rightarrow \mathbb{R}}\) is the Hamiltonian function.

Poisson systems:

These are dynamical systems whose vector field X H can be described through a Poisson structure (Poisson brackets ) defined on the ring of differentiable functions on a given manifold that is not necessarily symplectic (see “Hamiltonian Equations”). Note that on any symplectic manifold there is a natural Poisson structure such that any Hamiltonian system admits a Poisson formulation, but the contrary is false. The Poisson formulation of the dynamics is a generalization of the Hamiltonian one.

A periodic orbit:

\({\phi(.)}\) is a solution of the equations of motion that repeats itself after a certain time \({T > 0}\) called a period, that is, \({\phi(t+T)=\phi(t)}\) for every t.

Poincaré section/map:

Given a periodic orbit \({\phi(.)}\) a Poincaré section is a hyperplane S intersecting the curve \({\{\phi(t)\colon t\in [0,T)\}}\) transversely. The associated Poincaré map Π maps neighborhoods of S into itself by following the orbit \({\phi(.)}\) (see Definition 10).

A Hamiltonian system with symmetry:

is a Hamiltonian system in which there is a group G acting on \({\mathcal{P}}\), i. e., there is a map \({\Phi\colon G\times\mathcal{P}\mapsto \mathcal{P}}\), with Φ preserving the Hamiltonian and the symplectic form.

Relative periodic orbit:

Let G be a symmetry group for the dynamics. A path \({\phi(.)}\) is a relative periodic orbit if solves the equations of motion and repeats itself up to a group action after a certain time \({T > 0}\), that is, \({\phi(t+T)=\Phi_g(\phi(t))}\) for every t and for some \({g\in G}\).

Continuation:

Continuation is a procedure based on the implicit function theorem (IFT) that allows one to extend the solution of an equation for different values of the parameters. Let \({f(x,\epsilon)=0}\) be an equation in \({x\in\mathbb{R}^n}\) where f is differentiable and \({\epsilon\geq 0}\) a parameter. Assume that \({f(x_0,0)=0}\); a curve \({x(\epsilon)}\) is called a continued solution if \({x(0)=x_0}\) and \({f(x(\epsilon),\epsilon)=0}\) for some \({\epsilon\geq 0}\). In general \({x(\epsilon)}\) exists whenever the IFT can be applied, that is, if \({D_x f(x,\epsilon)}\) is invertible at (\({x_0,0}\)).

Liapunov–Schmidt reduction:

Let f be a function on a Banach space. Liapunov–Schmidt reduction is a procedure that allows one to study \({f(x,\epsilon)=0}\) under the condition that the kernel of Df is not empty but it is finite‐dimensional.

Variational principles:

The principles which aim to translate the problem of solving the equations of motion of a dynamical system (e. g., Hamiltonian systems) into the problem of finding the critical points of certain functionals defined on spaces of all possible trajectories of the given system.

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Acknowledgments

The author would like to thank Heinz Hanßmann for his critical and thorough reading of the manuscript and Ferdinand Verhulst for his useful suggestions.

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Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Warwick, UK

    Luca Sbano

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  1. Luca Sbano
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Sbano, L. (2012). Periodic Orbits of Hamiltonian Systems. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_74

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