Sharkovskii theorem for infinite dimensional dynamical systems

TL;DR

This work extends Sharkovskii’s ordering to infinite-dimensional dynamical systems by leveraging covering-relations methods that mimic one-dimensional dynamics. A general theorem shows that if a compact map on a Banach space close to a 1D map has an $n$-periodic orbit, then it possesses all periods $m$ with $n \triangleleft m$ in Sharkovskii order. The framework is instantiated for Delay Differential Equations, proving that small delayed perturbations of a chaotic ODE preserve the Sharkovskii property, with a detailed computer-assisted proof for a Rössler-like DDE demonstrating the existence of periodic orbits of all natural periods. The approach relies on horizontal h-sets with tails and an infinite-dimensional Itinerary Lemma, enabling rigorous verification of covering relations and periodic orbits via CAPD-based tools. The results indicate a broad, computationally verifiable pathway to establishing complex dynamics in infinite-dimensional systems such as DDEs and potentially PDEs.

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.

Figures (11)

• Figure 1: Horizontal self-covering $N \overset{f}{\Longrightarrow} N$. The image $f(N)$ is homotopically equivalent to the image $A(N)$ through a linear map $A(p,q)=(ap,0)$.
• Figure 2: Horizontal self-covering $N \overset{f}{\Longrightarrow} N$ in $\mathbb{R}^3$. Note that the 'right edge' $N^r$ (blue) is mapped to the right of the h-set and similarly the left edge $N^l$ (red) is mapped to the left.
• Figure 3: An example of a self-covering relation $N \stackrel{P}{\Longrightarrow} N$ on an h-set with tail $N = (N_1, |N_2|)$, $d_N = 2$. The head $N_1$ is the rectangle in the front, with red and blue sides (the exit and entrance set, respectively). The tail $|N_2|$ is closed and convex in a potentially infinite dimensional space (yellow dimension, the 'depth' of the picture).
• Figure 4: Illustration of assumptions of Theorem \ref{['thm:sh-manyD']}: $F(G)\subset\mathop{\mathrm{int}}\nolimits G$, $F(C_{i})\subset\mathop{\mathrm{int}}\nolimits C_{i+1}$, $i=1,2,3,4$
• Figure 5: The attracting $3$-periodic orbit for the system \ref{['eq:rossler525']}

Theorems & Definitions (46)

• Theorem 1: Sharkovskii, ShU
• Definition 1: (Proper) covering relation between intervals
• Theorem 2: Itinerary Lemma
• Definition 2: burns, Lemma 2.6
• Remark 3: burns, Lemma 2.6
• Theorem 4: GZ2022
• Definition 3: Definition 1 in GZ, for the case $u_{N}=1$
• Definition 4: cf. Definition 2 in GZ
• Remark 5
• Theorem 6: PZszarI, more generally GZ