Sharkovskii theorem for infinite dimensional dynamical systems

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.