Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates

TL;DR

This paper analyzes the number and stability of equilibria in a complex nonlinear random system with heterogeneous relaxation rates, revealing phase transitions and universal behaviors using advanced mathematical techniques.

Contribution

It extends previous models by deriving exact analytical results for systems with a spectrum of relaxation rates, revealing new phase transitions and universal properties.

Findings

Identifies a topology trivialisation transition from many to a single equilibrium.

Shows a phase with exponentially many stable equilibria depending on the gradient component.

Finds the complexity behavior at transitions depends only on small relaxation rate spectrum details.

Abstract

We consider a nonlinear autonomous random dynamical system of $N$ degrees of freedom coupled by Gaussian random interactions and characterized by a continuous spectrum $n_{\mu}(\lambda)$ of real positive relaxation rates. Using Kac-Rice formalism, the computation of annealed complexities (both of stable equilibria and of all types of equilibria) is reduced to evaluating the averages involving the modulus of the determinant of the random Jacobian matrix. In the limit of large system $N\gg 1$ we derive exact analytical results for the complexities for short-range correlated coupling fields, extending results previously obtained for the "homogeneous" relaxation spectrum characterised by a single relaxation rate. We show the emergence of a "topology trivialisation" transition from a complex phase with exponentially many equilibria to a simple phase with a single equilibrium as the magnitude of the random field is decreased. Within the complex phase the complexity of stable equilibria undergoes an additional transition from a phase with exponentially small probability to find a single equilibrium to a phase with exponentially many stable equilibria as the fraction of gradient component of the field is increased. The behaviour of the complexity at the transition is found only to depend on the small $\lambda$ behaviour of the spectrum of relaxation rates $n_{\mu}(\lambda)$ and thus conjectured to be universal. We also provide some insights into a counting problem motivated by a paper by B. Spivak and A. Zyuzin of 2004 about wave scattering in a disordered nonlinear medium.