Roman Srzednicki (Jagiellonian University)

TITLE:

On geometric detection of periodic solutions and chaotic dynamics of ordinary differential equations

ABSTRACT: Following ideas of Wazewski and Conley we introduce the notion of isolating segment, a set of a special form in the extended phase space of a non-autonomous ordinary differential equation. Under an additional assumption on time-periodicity of the segment, we present a theorem on the fixed point index of the Poincare map associated to the equation. As consequences, we get a results on the existence of periodic solutions and results on the existence of symbolic dynamics of the Poincare map. We illustrate those results by examples of planar polynomial equations with periodic coefficients.