Prevalent Behavior of Smooth Strongly Monotone Discrete-Time Dynamical Systems

TL;DR

This paper demonstrates that in smooth strongly monotone discrete-time dynamical systems, convergence to stable cycles is common, and applies these findings to certain parabolic equations, showing prevalent convergence to periodic solutions.

Contribution

It establishes the prevalence of convergence to periodic solutions in smooth strongly monotone systems and applies these results to classes of parabolic equations with specific boundary conditions.

Findings

Convergence to linearly stable cycles is prevalent in measure-theoretic sense.

Prevalence of convergence to spatially homogeneous periodic solutions in convex domains.

Prevalence of convergence to radially symmetric periodic solutions in symmetric domains.

Abstract

For C1-smooth strongly monotone discrete-time dynamical systems, it is shown that ``convergence to linearly stable cycles" is a prevalent asymptotic behavior in the measuretheoretic sense. The results are then applied to classes of time-periodic parabolic equations and give new results on prevalence of convergence to periodic solutions. In particular, for equations with Neumann boundary conditions on convex domains, we show the prevalence of the set of initial conditions corresponding to the solutions that converge to spatiallyhomogeneous periodic solutions. While, for equations on radially symmetric domains, we obtain the prevalence of the set of initial values corresponding to solutions that are asymptotic to radially symmetric periodic solutions.