Many unstable fixed points, and chaotic dynamics: an high-dimensional example

Complex systems tend to exhibit out-of-equilibrium dynamics over a broad range of timescales. A key theory challenge is to understand the features of this out-of-equilibrium behavior from the properties of the attractors of the system’s dynamical equations. Mean-field theories of spin glass dynamics offer elegant examples, linking phenomena such as aging to the properties of special families of stationary points of the underlying free-energy landscape, that attract the dynamics at large times. However, these insights apply mainly to systems to which one can associate such landscapes. It is an open question how to extend these ideas to high-dimensional non-conservative systems (such as biological neural networks, large ecosystems) whose dynamics is not landscape optimization. I will present a simple model of a high-dimensional system with non-reciprocal interactions whose out-of-equilibrium, chaotic dynamics can be characterized analytically at long times. I will discuss its dynamical phase diagram and compare it to the statistical distribution of the many, unstable fixed points of the dynamical equations. This comparison challenges the idea that chaotic dynamics in non-conservative settings can be understood from fixed points alone, at least from the typical ones.

The results are presented in arXiv:2503.20908