Part of the International Centre for Mechanical Sciences book series (CISM, volume 371)
On Geometric Detection of Periodic Solutions and Chaos
In this note we consider a differential equationwhere f : ℝ × M → TM is a continuous time-dependent vector-field on a smooth manifold M. We assume that the Cauchy problemassociated to (1) has the uniqueness property. Fix some T > 0 and let the map f (·, x) be T-periodic for every x ∈ M. Examples of equations of that form arise, for example, in mechanical systems perturbed by periodic forces. One of the most important problems referred to (1) in that case is to determine the existence of its periodic solutions with the period being a multiplicity of T. Classical methods related to that problem are presented in [RM]. The purpose of this note is to describe a geometric approach to it, introduced in the paper [S2] (see also [S1] and [S4]). Actually, we present (without complete proofs) its minor modification and applications to detecting chaotic dynamics. Full exposition of the topics concerning chaos which are presented here is contained in the papers [S5] and [SW].
KeywordsPeriodic Solution Chaotic Dynamic Periodic Point Lefschetz Number Conley Index
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- [C]C. C. Conley, Isolated Invariant Sets and the Morse Index, CBMS vol. 38, Amer. Math. Soc., Providence, RI 1978.Google Scholar
- [Ma]J. Mawhin, Periodic solutions of some complex-valued differential equations with periodic coefficients, in: G. Lumer et al (editors), Partial Differential Equations. Contributions to the conference held in Han-sur-Lesse, Math. Res. 82, Akademie-Verlag, Berlin 1993, 226–234.Google Scholar
- [S1]R. Srzednicki, A geometric method for the periodic problem in ordinary differential equations, Séminaire d’Analyse Moderne, No. 22, Université de Sherbrooke 1992, 1–110.Google Scholar
- [S4]R. Srzednicki, A geometric method for the periodic problem, in: V. Lakshmikantham (editor), Proceedings of the First World Congress of Nonlinear Analysts, Tampa, Florida, August 19–26, 1992, Walter de Gruyter, Berlin, New York 1996, 549–560.Google Scholar
- [S5]R. Srzednicki, On chaotic dynamics inside isolating blocks, in preparation.Google Scholar
- [SW]R. Srzednicki, K. Wojcik, A geometric method for detecting chaotic dynamics, preprint, Jagiellonian University, Krakow 1996.Google Scholar
- [Wo]K. Wójcik, On detecting periodic solutions and chaos in ODEs, in preparation.Google Scholar
© Springer-Verlag Wien 1996