Stability and convergence in discrete convex monotone dynamical systems

TL;DR

This paper investigates the stability and convergence properties of discrete convex monotone dynamical systems, introducing tangential stability and providing criteria for fixed points and periodic orbits.

Contribution

It introduces the concept of tangential stability, analyzes its structure, and offers new criteria for the existence and uniqueness of stable fixed points and periodic orbits.

Findings

Set of tangentially stable fixed points forms a convex inf-semilattice

Criteria for the existence of a unique tangentially stable fixed point

Conditions for global convergence to periodic orbits

Abstract

We study the stable behaviour of discrete dynamical systems where the map is convex and monotone with respect to the standard positive cone. The notion of tangential stability for fixed points and periodic points is introduced, which is weaker than Lyapunov stability. Among others we show that the set of tangentially stable fixed points is isomorphic to a convex inf-semilattice, and a criterion is given for the existence of a unique tangentially stable fixed point. We also show that periods of tangentially stable periodic points are orders of permutations on $n$ letters, where $n$ is the dimension of the underlying space, and a sufficient condition for global convergence to periodic orbits is presented.