Sharpened dynamics alternative and its $C^1$-robustness for strongly monotone discrete dynamical systems

TL;DR

This paper establishes a refined dichotomy for strongly monotone discrete dynamical systems, showing that orbits either diverge or converge to stable cycles with bounded periods, and demonstrates this property’s robustness under small perturbations.

Contribution

The paper introduces a sharpened dynamics alternative for $C^1$-smooth strongly monotone discrete systems and proves its $C^1$-robustness, extending the understanding of convergence to cycles.

Findings

Orbits are either unstable or asymptotic to stable cycles with bounded period.

The sharpened dynamics alternative is robust under $C^1$-perturbations.

Results imply generic convergence to periodic solutions with bounded periods.

Abstract

For strongly monotone dynamical systems, the dynamics alternative for smooth discrete-time systems turns out to be a perfect analogy of the celebrated Hirsch's limit-set dichotomy for continuous-time semiflows. In this paper, we first present a sharpened dynamics alternative for $C^1$-smooth strongly monotone discrete-time dissipative system $\{F_0^n\}_{n\in \mathbb{N}}$ (with an attractor $A$), which concludes that there is a positive integer $m$ such that any orbit is either manifestly unstable; or asymptotic to a linearly stable cycle whose minimal period is bounded by $m$. Furthermore, we show the $C^1$-robustness of the sharpened dynamics alternative, that is, for any $C^1$-perturbed system $\{F_\epsilon^n\}_{n\in \mathbb{N}}$ ($F_\epsilon$ not necessarily monotone), any orbit initiated nearby $A$ will admit the sharpened dynamics alternative with the same $m$. The improved generic convergence to cycles for the $C^1$-system $\{F_0^n\}_{n\in \mathbb{N}}$, as well as for the perturbed system $\{F_\epsilon^n\}_{n\in \mathbb{N}}$, is thus obtained as by-products of the sharpened dynamics alternative and its $C^1$-robustness. The results are applied to nonlocal $C^1$-perturbations of a time-periodic parabolic equations and give typical convergence to periodic solutions whose minimal periods are uniformly bounded.