Stability of Fixed Points for Nonlinear Selfconsistent Transfer Operators via Cone Contractions

TL;DR

This paper analyzes the stability of fixed points for nonlinear self-consistent transfer operators using cone contraction methods, providing explicit estimates and examples of multiple stable fixed points in coupled map systems.

Contribution

It introduces a new approach based on cone contractions to study the stability of fixed points of self-consistent transfer operators, including explicit stability conditions and examples.

Findings

Identifies conditions for stability of fixed points in STOs

Provides explicit estimates for stability analysis

Shows existence of multiple stable fixed points in coupled maps

Abstract

In this paper we investigate the action of self-consistent transfer operators (STOs) on Birkhoff cones and give sufficient conditions for stability of their fixed points. Our approach relies on the order preservation properties of STOs that can be established via the study of their differential. We focus on the study of STOs arising from strongly coupled maps both deterministic and noisy. Our approach allows for explicit estimates that we use to give examples of STOs with multiple stable fixed points some of which are shown to be far from the asymptotic behaviour of the corresponding system of finite coupled maps and give information only on long transients for the finite dimensional system.