A Geometric Method for Detecting Chaotic Dynamics

Abstract

A new method of detection of chaos in dynamical systems generated by time-periodic nonautonomous differential equations is presented. It is based on the existence of some sets (called periodic isolating segments) in the extended phase space, satisfying some topological conditions. By chaos we mean the existence of a compact invariant set such that the Poincare map is semiconjugated to the shift on two symbols and the counterimage (by the semiconjugacy) of any periodic point in the shift contains a periodic point of the Poincare map. As an application we prove that the planar equation z* =(1+e i,t |z| 2 ) zÄ generates chaotic dynamics provided 0<, 1Â288.1997 Academic Press

Superlinear Indefinite Equations on the Real Line and Chaotic Dynamics

“…More recently, contributions have been made by, among others, Battelli and Palmer [3], Srzednicki [17], and Srzednicki and W ! o ojcik [18]; in [3] the existence of a transversal homoclinic orbit for the Poincar! e e map associated to a Duffing equation is studied and in [18] some geometric conditions for detection of chaos are given (see also Remark 3.10).…”

Isolating Segments, Fixed Point Index, and Symbolic Dynamics: III. Applications

“…For (1) using Srzednicki-Wojcik gives symbolic dynamics on two symbols only for all n (see [SW,W1,W2]), whereas we prove here that we have symbolic dynamics on ðn þ 2Þ symbols.…”

Periodic segments and Nielsen numbers

“…This result has many applications in detecting periodic solutions and chaos in nonautonomous periodic differential equations (see [9], [10], [11], [12], [13]). We will need some notions related to W .…”

“…Introduction. In [9] Roman Srzednicki introduced the geometric method for detecting periodic solutions in nonautonomous periodic differential equations based on the notion of periodic isolating blocks (or periodic isolating segments considered in [11], [13]). The method is based on the Lefschetz Fixed Point Theorem and the Ważewski Retract Theorem.…”