Sharkovskiis theorem under small random perturbations

TL;DR

The paper tackles extending Sharkovskiĭ's theorem to small random perturbations of interval maps by employing the random Conley index, yielding a topological forcing mechanism that detects random periodic points with precise minimal periods. It defines a full random dynamical systems framework, introduces random periodic points and $(\delta,k)$-random periodic orbits, and proves forcing and realisation results for finite Sharkovskiĭ tails, avoiding the period-doubling ambiguity common in measure-theoretic approaches. Central to the approach is the persistence of a nontrivial Conley index under small random perturbations, guaranteed by a random continuation property and the random Ważewski principle, which together produce random periodic structures corresponding to prescribed periods. The results are illustrated with explicit constructions for random perturbations of the tent and logistic maps, demonstrating the practical realisation of targeted Sharkovskiĭ tails in the stochastic setting. Overall, the work provides a robust topological framework for understanding how deterministic periodic structures persist and manifest as random periodic dynamics under small noise, with potential implications for predicting and controlling intermittency and periodicity in noisy one-dimensional systems.

Abstract

We establish a Sharkovskii-type theorem for a class of discrete random dynamical systems via the random Conley index. Using the continuation property of the Conley index, we extend classical forcing results to random systems obtained from small random perturbations of one-dimensional maps. In contrast to earlier measure-theoretic results, which are typically subject to an inherent period-doubling ambiguity (realizing period $n$ or $2n$), our topological approach allows us to detect random periodic points and orbits with precise minimal periods. This yields realisation results for arbitrary finite tails of the Sharkovskii ordering. These results are illustrated by constructing random periodic orbits for perturbed versions of the tent map and the logistic map.

Figures (4)

• Figure 1: Geometric illustration of an $(\delta, k)$-random periodic orbit. The trajectory starts at $x_0$ inside the neighborhood $B_\delta$, performs an excursion outside the neighborhood for intermediate steps $l \in \{1, \dots, k-1\}$, and returns to $B_\delta$ at step $k$. This defines a form of periodicity with spatial tolerance $\delta$.
• Figure 2: The truncated tent map $T_h$: the thick graph is $T_h$, while the gray graph is the standard tent map and the diagonal.
• Figure 3: Random iteration of asymmetric tent maps $T_\gamma$ with common peak height $1$. The coloured graphs represent several maps $T_\gamma$ with different peak locations $x = \tfrac{1}{2} + \gamma$, and the red staircase shows a sample trajectory in the $(x_n,x_{n+1})$ plane obtained by randomly switching between these maps.
• Figure 4: Graphs of the logistic maps $f_c(x)=c\,x(1-x)$ on $[0,1]$ for the parameter values $c=0.8,0.9,1.0,1.1$ and $1.2$.

Theorems & Definitions (44)

• Theorem 1: Sharkovskiĭ's Forcing Theorem sharkovskiui1995coexistence
• Theorem 2: Sharkovskiĭ's Realisation Theorem sharkovskiui1995coexistence
• Definition 3: klunger2001periodicity
• Definition 4: klunger2001periodicity
• Definition 5: $(\delta,k)$--random periodic orbit
• Definition 6: Finite Sharkovskiĭ tail
• Definition 7
• Definition 8
• Theorem 9: Random Sharkovskiĭ forcing for finite tails
• Theorem 10: Random realisation of finite Sharkovskii tails