Sharkovskii theorem for infinite dimensional dynamical systems

TL;DR

This paper extends Sharkovskii's theorem to certain infinite-dimensional dynamical systems, showing the existence of all periodic orbits of specific periods when a system has a particular periodic orbit, with applications to delay differential equations.

Contribution

It introduces a Sharkovskii-type theorem for infinite-dimensional systems close to one-dimensional maps, verified via computer, with applications to delay differential equations.

Findings

Existence of all periodic orbits of certain periods in infinite-dimensional systems.

Application to delay differential equations demonstrating the theorem.

Computer-assisted verification of assumptions.

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 R\"ossler ODE system perturbed by a delayed term and we show that it retains periodic orbits of all natural periods for fixed values of parameters.