Non Linear Analysis and Boundary Value Problems for Ordinary Differential Equations pp 197-209 | Cite as

# On Geometric Detection of Periodic Solutions and Chaos

Chapter

## Abstract

In this note we consider a differential equation
where
associated to (1) has the uniqueness property. Fix some

(1)

*f*: ℝ ×*M*→*TM*is a continuous time-dependent vector-field on a smooth manifold*M*. We assume that the Cauchy problem(2)

*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].## Keywords

Periodic 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