Dynamical Transitions in Large Systems of Mean Field-Coupled Landau-Stuart Oscillators: Extensive Chaos and Clumped States

TL;DR

This paper investigates the complex dynamical behaviors of large systems of coupled Landau-Stuart oscillators, revealing extensive chaos with linear scaling of fractal dimension and Lyapunov exponents, and characterizing explosive transitions between clumped and chaotic states.

Contribution

It demonstrates the extensive nature of chaos in large oscillator systems and analyzes the mechanism behind explosive transitions between ordered and chaotic states.

Findings

Chaos scales linearly with system size

Explosive transitions are discontinuous and involve marginally stable clumped states

Fractal structure analyzed using Kaplan-Yorke formula

Abstract

In this paper, we study dynamical systems in which a large number $N$ of identical Landau-Stuart oscillators are globally coupled via a mean-field. Previously, it has been observed that this type of system can exhibit a variety of different dynamical behaviors including clumped states in which each oscillator is in one of a small number of groups for which all oscillators in each group have the same state which is different from group to group, as well as situations in which all oscillators have different states and the macroscopic dynamics of the mean field is chaotic. We argue that this second type of behavior is $^{\backprime}$extensive$^{\prime}$ in the sense that the chaotic attractor in the full phase space of the system has a fractal dimension that scales linearly with $N$ and that the number of positive Lyapunov exponents of the attractor also scales with linearly $N$. An important focus of this paper is the transition between clumped states and extensive chaos as the system is subjected to slow adiabatic parameter change. We observe explosive (i.e., discontinuous) transitions between the clumped states (which correspond to low dimensional dynamics) and the extensively chaotic states. Furthermore, examining the clumped state, as the system approaches the explosive transition to extensive chaos, we find that the oscillator population distribution between the clumps continually evolves so that the clumped state is always marginally stable. This behavior is used to reveal the mechanism of the explosive transition. We also apply the Kaplan-Yorke formula to study the fractal structure of the extensively chaotic attractors.