High-dimensional dynamical systems: co-existence of attractors, phase transitions, maximal Lyapunov exponent and response to periodic drive

TL;DR

This paper investigates high-dimensional random dynamical systems, analyzing their attractors, phase transitions, Lyapunov exponents, and response to periodic forcing, revealing insights into their complex behavior and potential applications in neural networks.

Contribution

It introduces a class of high-dimensional systems with co-existing attractors, explicitly computes Lyapunov exponents, and studies their response to periodic stimuli, simplifying analysis compared to traditional models.

Findings

Attractors can co-exist in certain phase diagram regions.

Maximal Lyapunov exponent can be explicitly computed in some cases.

Systems act as frequency filters, showing synchronization only in specific regions.

Abstract

We study the dynamical properties of a broad class of high-dimensional random dynamical systems exhibiting chaotic as well as fixed point and periodic attractors. We consider cases in which attractors can co-exists in some regions of the phase diagrams and we characterize their nature by computing the maximal Lyapunov exponent. For a specific choice of the dynamical system we show that this quantity can be computed explicitly in the whole chaotic phase due to an underlying integrability of a properly defined Schr\"odinger problem. Furthermore, we consider the response of this dynamical systems to periodic perturbations. We show that these dynamical systems act as filters in the frequency-amplitude spectrum of the periodic forcing: only in some regions of the frequency-amplitude plane the periodic forcing leads to a synchronization of the dynamics. All in all, the results that we present mirror closely the ones observed in the past forty years in the study of standard models of random recurrent neural networks. However, the dynamical systems that we consider are easier to study and we believe that this may be an advantage if one wants to go beyond random dynamical systems and consider specific training strategies.