Chaos in high-dimensional dynamical systems with tunable non-reciprocity

TL;DR

This paper explores how partial non-reciprocity in high-dimensional dynamical systems influences chaos, revealing that non-reciprocal interactions can induce and enhance chaotic behavior.

Contribution

It demonstrates that systems with partially symmetric interactions can exhibit chaos, with the maximal Lyapunov exponent varying non-monotonically with non-reciprocity.

Findings

Chaotic attractors occur for all non-reciprocal interaction parameters.

Maximal Lyapunov exponent varies non-monotonically with non-reciprocity.

Non-reciprocal interactions can increase chaos even in systems with gradient-like forces.

Abstract

High-dimensional dynamical systems of interacting degrees of freedom are ubiquitous in the study of complex systems. When the directed interactions are totally uncorrelated, sufficiently strong and non-linear, many of these systems exhibit a chaotic attractor characterized by a positive maximal Lyapunov exponent (MLE). On the contrary, when the interactions are completely symmetric, the dynamics takes the form of a gradient descent on a carefully defined cost function, and it exhibits slow dynamics and aging. In this work, we consider the intermediate case in which the interactions are partially symmetric, with a parameter {\alpha} tuning the degree of non-reciprocity. We show that for any value of {\alpha} for which the corresponding system has non-reciprocal interactions, the dynamics lands on a chaotic attractor. Correspondingly, the MLE is a non-monotonous function of the degree of non-reciprocity. This implies that conservative forcing deriving from the gradient field of a rough energy landscape can make the system more chaotic.