Sharkovskiis theorem under small random perturbations

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.

💡 Research Summary

This paper extends the classical Sharkovskiĭ theorem to discrete‑time random dynamical systems (RDS) generated by small random perturbations of one‑dimensional interval maps. The authors employ the random Conley index, a topological invariant that enjoys a continuation property analogous to its deterministic counterpart, to transfer forcing relations from the deterministic setting to the stochastic one.

After recalling the Sharkovskiĭ ordering and the classical forcing and realisation theorems, the authors introduce three notions of random periodicity. A “random periodic point” (Definition 3) is a measurable map (x:\Omega\to\mathbb R) satisfying (\phi(k,\omega)x(\omega)=x(\theta^{k}\omega)) almost surely, with minimal period defined by the absence of shorter repetitions. A “random periodic orbit” (Definition 4) is a random invariant set whose fibre contains exactly (k) points almost surely. The novel concept of a “((\delta,k))-random periodic orbit” (Definition 5) adds a geometric constraint: each fibre of the orbit is (\delta)-small in diameter and distinct fibres are separated by more than (\delta). This refinement allows the detection of precise spatial localisation, which earlier definitions lack.

The paper then focuses on finite Sharkovskiĭ tails (T\subset\mathbb N), i.e. finite initial segments of the Sharkovskiĭ ordering. For a deterministic map (f) belonging to the class (H_T) (Definition 6), every periodic orbit of period (k\in T) is hyperbolic and lies in the interior of the interval. Around each such orbit the authors choose disjoint neighbourhoods and define a class (R_1^{\varepsilon}(f)) of random maps that are uniformly (C^0)‑close to (f) on the whole interval and uniformly (C^1)‑close on the chosen neighbourhoods.

Theorem 9 (Random Sharkovskiĭ forcing for finite tails) states that for any finite tail (T) and any (f\in H_T) there exists (\varepsilon>0) such that every RDS generated by a map (\varphi\in R_1^{\varepsilon}(f)) with an ergodic base (\theta) possesses:

1. a random periodic point of minimal period (k) for every (k\in T);
2. if (\theta^{k}) is ergodic for all (k), a random periodic orbit of minimal period (k) for every (k\in T);
3. a ((\delta,k))-random periodic orbit for each (k\in T) with some (\delta>0).

Thus the deterministic forcing relation “(p\prec q) implies existence of period (q)” survives unchanged under sufficiently small stochastic perturbations, and the minimal periods are identified without the usual period‑doubling ambiguity.

Theorem 10 (Random realisation of finite tails) provides the converse: for any finite tail (T) one can construct non‑trivial RDS (with various ergodicity assumptions) whose sets of minimal periods of random periodic points, random periodic orbits, or ((\delta,k))-random periodic orbits exactly contain (T).

The proofs rest on the continuation property of the random Conley index. Hyperbolic periodic points give rise to isolating neighbourhoods with well‑defined index pairs ((N,L)). Small random perturbations preserve these index pairs, guaranteeing the existence of invariant random sets inside the neighbourhoods. By projecting these invariant sets onto the fibres one obtains the three types of random periodic structures. The geometric ((\delta,k)) condition follows from controlling the diameter of the fibres inside the isolating neighbourhoods.

Section 5 illustrates the abstract results with two concrete families. For the tent map (T(x)=1-2|x-1/2|) and the logistic map (L_r(x)=rx(1-x)) (with parameters chosen to have rich periodic structure), the authors add small uniform or Gaussian noise respectively. Numerical simulations confirm that the perturbed systems exhibit random periodic points and orbits of all periods prescribed by chosen finite tails, and that the ((\delta,k)) separation is observable.

The paper positions its contributions relative to earlier works. Andres