Nonlinear Analogue of the May-Wigner Instability Transition

TL;DR

This paper investigates a high-dimensional nonlinear system with Gaussian coupling, revealing an abrupt transition from a simple to a complex equilibrium landscape as coupling strength increases, offering a global perspective on the May-Wigner instability.

Contribution

It introduces a nonlinear analogue of the May-Wigner transition, showing how increasing coupling leads to a topological change in the system's equilibrium structure.

Findings

Transition from a single to exponentially many equilibria

Most equilibria are unstable in the complex regime

Provides a global topological view of the May-Wigner transition

Abstract

We study a system of $N\gg 1$ degrees of freedom coupled via a smooth homogeneous Gaussian vector field with both gradient and divergence-free components. In the absence of coupling, the system is exponentially relaxing to an equilibrium with rate $\mu$. We show that, while increasing the ratio of the coupling strength to the relaxation rate, the system experiences an abrupt transition from a topologically trivial phase portrait with a single equilibrium into a topologically non-trivial regime characterised by an exponential number of equilibria, the vast majority of which are expected to be unstable. It is suggested that this picture provides a global view on the nature of the May-Wigner instability transition originally discovered by local linear stability analysis.