High-dimensional chaos is more than just wandering around unstable equilibria.

Complex systems tend to equilibrate slowly, exhibiting 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 of this, linking phenomena such as aging to the properties of the stationary points of the underlying free-energy landscape (which are attractors of the dynamics). However, these insights apply mainly to conservative systems. In recent years, there has been growing interest in extending these ideas to high-dimensional non-conservative systems, motivated by neural networks and theoretical ecology. In this talk, I will present a simple model of a high-dimensional system with non-reciprocal interactions, whose chaotic, out-of-equilibrium dynamics can be analyzed analytically at long times. I will discuss its dynamical phase diagram and compare it to the statistical distribution of the many, unstable equilibria of the dynamical equations. This comparison challenges the common assumption that chaotic dynamics in non-conservative settings can be understood from equilibria alone. The results rely on a combination of two analytical techniques, Dynamical Mean-Field Theory and the Kac-Rice formalism, and are presented in arXiv:2503.20908.