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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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© Springer-Verlag Wien 1996