Topology trivialization transition in random non-gradient autonomous ODE's on a sphere

TL;DR

This paper introduces a new method to calculate the average number of equilibrium points in high-dimensional random autonomous ODEs on a sphere, revealing a phase transition from many equilibria to just two as randomness increases.

Contribution

It proposes a novel approach integrating Lagrange multipliers into the Kac-Rice framework for analyzing equilibrium counts in nonrelaxational dynamics.

Findings

Confirmed topology trivialization in nonrelaxational systems

Identified a phase transition in the number of equilibria

Provided a new computational method for equilibrium analysis

Abstract

We calculate the mean total number of equilibrium points in a system of $N$ random autonomous ODE's introduced by Cugliandolo et al. in 1997 to describe non-relaxational glassy dynamics on the high-dimensional sphere. In doing it we suggest a new approach which allows such a calculation to be done most straightforwardly, and is based on efficiently incorporating the Langrange multiplier into the Kac-Rice framework. Analysing the asymptotic behaviour for large $N$ we confirm that the phenomenon of 'topology trivialization' revealed earlier for other systems holds also in the present framework with nonrelaxational dynamics. Namely, by increasing the variance of the random 'magnetic field' term in dynamical equations we find a 'phase transition' from the exponentially abundant number of equilibria down to just two equilibria. Classifying the equilibria in the nontrivial phase by stability remains an open problem.