On Geometric Detection of Periodic Solutions and Chaos

  • R. Srzednicki
Part of the International Centre for Mechanical Sciences book series (CISM, volume 371)


In this note we consider a differential equation
where f : ℝ × MTM is a continuous time-dependent vector-field on a smooth manifold M. We assume that the Cauchy problem
associated to (1) has the uniqueness property. Fix some T > 0 and let the map f (·, x) be T-periodic for every xM. 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].


Periodic Solution Chaotic Dynamic Periodic Point Lefschetz Number Conley Index 
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Copyright information

© Springer-Verlag Wien 1996

Authors and Affiliations

  • R. Srzednicki
    • 1
  1. 1.Jagiellonian UniversityCracowPoland

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