Chaos in high-dimensional dynamical systems

TL;DR

This paper investigates how the likelihood of chaos in high-dimensional dissipative dynamical systems increases with dimension, revealing universal scaling laws and providing analytical explanations for the transition to chaos.

Contribution

It demonstrates that the probability of chaos approaches one as dimension increases and uncovers universal scaling behaviors independent of specific coefficients.

Findings

Probability of chaos increases with dimension, reaching near certainty at d~50.

Universal scaling laws govern the invariant measure and Lyapunov exponents in high dimensions.

Analytical explanations for chaos probability and scaling are provided.

Abstract

For general dissipative dynamical systems we study what fraction of solutions exhibit chaotic behavior depending on the dimensionality $d$ of the phase space. We find that a system of $d$ globally coupled ODE's with quadratic and cubic non-linearities with random coefficients and initial conditions, the probability of a trajectory to be chaotic increases universally from $\sim 10^{-5} - 10^{-4}$ for $d=3$ to essentially one for $d\sim 50$. In the limit of large $d$, the invariant measure of the dynamical systems exhibits universal scaling that depends on the degree of non-linearity but does not depend on the choice of coefficients, and the largest Lyapunov exponent converges to a universal scaling limit. Using statistical arguments, we provide analytical explanations for the observed scaling and for the probability of chaos.