Sharkovskii theorem for infinite dimensional dynamical systems

2024/11/28 by Gierzkiewicz, Anna, Szczelina, Robert
#34K13 #34K23 #34K27 #37B10 #65G20 #Dynamical Systems (math.DS) #FOS: Mathematics #G.1.7

paper · doi:10.48550/arxiv.2411.19190

Abstract

We present an adaptation of a relatively simple topological argument to show the existence of many periodic orbits in an infinite dimensional dynamical system, provided that the system is close to a one-dimensional map in a certain sense. Namely, we prove a Sharkovskii-type theorem: if the system has a periodic orbit of basic period m, then it must have all periodic orbits of periods n \triangleright m, for n preceding m in Sharkovskii ordering. The assumptions of the theorem can be verified with computer assistance, and we demonstrate the application of such an argument in the case of Delay Differential Equations (DDEs): we consider the Rössler ODE system perturbed by a delayed term and we show that it retains periodic orbits of all natural periods for fixed values of parameters.

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