# May-Wigner transition in large random dynamical systems

## TL;DR

This paper investigates the stability of large, complex, nonlinear random dynamical systems, revealing a phase transition similar to the May-Wigner transition in linear models, using stochastic analysis of Lyapunov exponents.

## Contribution

It introduces a stochastic framework for analyzing stability in high-dimensional nonlinear systems, extending the May-Wigner transition concept beyond linear models.

## Key findings

- Identification of a stability-complexity phase transition in nonlinear systems
- Derivation of a stochastic model for finite-time Lyapunov exponents
- Comparison with existing linear models and known results

## Abstract

We consider stability in a class of random non-linear dynamical systems characterised by a relaxation rate together with a Gaussian random vector field which is white-in-time and spatial homogeneous and isotropic. We will show that in the limit of large dimension there is a stability-complexity phase transition analogue to the so-called May-Wigner transition known from linear models. Our approach uses an explicit derivation of a stochastic description of the finite-time Lyapunov exponents. These exponents are given as a system of coupled Brownian motions with hyperbolic repulsion called geometric Dyson Brownian motions. We compare our results with known models from the literature.