Nonreciprocal interactions and high-dimensional chaos: Comparing dynamics and statistics of equilibria in a solvable class of models.

Abstract

We investigate a class of models of high-dimensional dynamics with all-to-all, random, and nonreciprocal interactions. We determine the dynamical phase diagram and show the emergence of chaotic dynamics. Correspondingly, we show that the dynamical equations exhibit a number of equilibria that is exponentially large in the system's dimensionality, all linearly unstable in the chaotic phases. Solving the effective equations governing the dynamics in the infinite-dimensional limit, we determine the typical properties (magnetization, overlap) of the configurations belonging to the attractor manifold. We show that these properties cannot be inferred from those of the equilibria, challenging the expectation that chaos can be understood purely in terms of the numerous unstable equilibria of the dynamical equations. We discuss the dependence of this scenario on the strength of nonreciprocity. The results are obtained combining analytical methods such as dynamical mean-field theory and the Kac-Rice formalism.