Counting equilibria of large complex systems by instability index

TL;DR

This paper analyzes large complex systems with random interactions, revealing phase transitions in equilibrium stability and abundance as interaction strength varies, highlighting the roles of gradient and solenoidal interactions.

Contribution

It introduces a framework to count and characterize equilibria in large nonlinear systems with mixed random interactions, identifying phase transitions in stability and abundance.

Findings

Transition from a single stable equilibrium to many unstable equilibria with increasing interactions.

Stable equilibria become exponentially abundant at high interaction strengths in purely solenoidal systems.

Calculated the proportion of equilibria with a fixed fraction of unstable directions.

Abstract

We consider a nonlinear autonomous system of $N\gg 1$ degrees of freedom randomly coupled by both relaxational ('gradient') and non-relaxational ('solenoidal') random interactions. We show that with increased interaction strength such systems generically undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically non-trivial regime of 'absolute instability' where equilibria are on average exponentially abundant, but typically all of them are unstable, unless the dynamics is purely gradient. When interactions increase even further the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal. We further calculate the mean proportion of equilibria which have a fixed fraction of unstable directions.