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Some Qualitative Properties of Lyapunov Exponents for Almost Periodic Hamiltonian Systems

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Abstract

The Lyapunov characteristic exponents play a crucial role in the description of the behavior of dynamical systems. They measure the average rate of convergence or divergence of orbits starting from nearby initial points. Therefore, they can be used to analyze the stability of limits sets and check sensitive dependence on initial conditions, that is, the presence of chaotic attractors. In this paper we explore some properties of Lyapunov exponents of autonomous almost periodic Hamiltonian systems. We study the behavior of the Lyapunov exponents under smooth conjugacy. We show that, due to almost periodicity, the Lyapunov exponents of the almost periodic Hamiltonian system are invariant under a smooth conjugacy. In addition, we obtain effective estimates of Lyapunov exponents by the mean value of the largest eigenvalue of the Lie Bracket \(\left[ J,H^{\prime \prime }\right] \). Finally, two numerical examples are given to indicate the validity of the obtained estimates.

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Acknowledgments

I would like to thank two anonymous reviewers for their careful reading and providing pertinent suggestions.

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  1. ISSATs and Laboratory of Mathematical Physics, Specials Functions and Applications, LR11ES35, Ecole Supérieure des Sciences et de Technologie de Hammam-Sousse, Université de Sousse, Sousse, Tunisia

    Farouk Chérif

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  1. Farouk Chérif
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Correspondence to Farouk Chérif.

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Chérif, F. Some Qualitative Properties of Lyapunov Exponents for Almost Periodic Hamiltonian Systems. Differ Equ Dyn Syst 23, 57–67 (2015). https://doi.org/10.1007/s12591-013-0180-8

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