Nonlinear analog of the complexity-stability transition in random dynamical systems: a replica calculation

TL;DR

This paper investigates a phase transition in high-dimensional dynamical systems with random forces, using replica calculations to analyze how the number of stationary points shifts from trivial to exponentially complex as randomness increases.

Contribution

It introduces a replica-based analytical approach to study the nonlinear analog of the complexity-stability transition in large random dynamical systems.

Findings

Identifies a topological trivialization phase transition.

Shows the number of stationary points shifts from one to exponential.

Provides a theoretical framework for analyzing complexity in dynamical systems.

Abstract

We consider large-dimensional dynamical systems involving a linear force and a random force comprising both potential and non-conservative contributions. Such systems are known to exhibit a topological trivialization phase transition as the strength of the random force is increased. This is reflected in the number of stationary points of the dynamical systems that transitions from one to an exponential in the number of degrees of freedom. We analyze this transition by means of a replica calculation.