# What drives transient behaviour in complex systems?

## TL;DR

This paper investigates the transient dynamics of complex systems modeled by non-linear ODEs, revealing new regimes and statistical distributions of transient trajectories through an extended May-Wigner random matrix framework.

## Contribution

It introduces a novel stable-transient regime, extends the May-Wigner model to include transient behavior, and derives exact distributions for transient trajectories using advanced random matrix theory.

## Key findings

- Identification of a stable-transient regime based on initial response.
- Exact calculation of transient trajectory distributions, including Gaussian and Tracy-Widom.
- Connection of transient behavior to eigenvector non-orthogonality and an extended May-Wigner model.

## Abstract

We study transient behaviour in the dynamics of complex systems described by a set of non-linear ODE's. Destabilizing nature of transient trajectories is discussed and its connection with the eigenvalue-based linearization procedure. The complexity is realized as a random matrix drawn from a modified May-Wigner model. Based on the initial response of the system, we identify a novel stable-transient regime. We calculate exact abundances of typical and extreme transient trajectories finding both Gaussian and Tracy-Widom distributions known in extreme value statistics. We identify degrees of freedom driving transient behaviour as connected to the eigenvectors and encoded in a non-orthogonality matrix $T_0$. We accordingly extend the May-Wigner model to contain a phase with typical transient trajectories present. An exact norm of the trajectory is obtained in the vanishing $T_0$ limit where it describes a normal matrix.