A Geometric Method for Detecting Chaotic Dynamics

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 Poincaré 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 Poincaré map. As an application we prove that the planar equation =(1+eiφt z2) zgenerates chaotic dynamics provided 0<φ 1/288.