Origin and scaling of chaos in weakly coupled phase oscillators

TL;DR

This paper investigates how the largest Lyapunov exponent scales with system size in large ensembles of weakly coupled phase oscillators, revealing dependence on frequency distribution regularity and finite size fluctuations.

Contribution

It provides a novel analysis of Lyapunov exponent scaling in oscillator ensembles, combining numerical simulations with approximate analytical arguments.

Findings

Lyapunov exponent scales as 1/N for regular frequency distributions

Logarithmic corrections appear with strong frequency fluctuations

Finite size fluctuations are key to positive Lyapunov exponents

Abstract

We discuss the behavior of the largest Lyapunov exponent $\lambda$ in the incoherent phase of large ensembles of heterogeneous, globally-coupled, phase oscillators. We show that the scaling with the system size $N$ depends on the details of the spacing distribution of the oscillator frequencies. For sufficiently regular distributions $\lambda \sim 1/N$, while for strong fluctuations of the frequency spacing, $\lambda \sim \ln N/N$ (the standard setup of independent identically distributed variables belongs to the latter class). In spite of the coupling being small for large $N$, the development of a rigorous perturbative theory is not obvious. In fact, our analysis relies on a combination of various types of numerical simulations together with approximate analytical arguments, based on a suitable stochastic approximation for the tangent space evolution. In fact, the very reason for $\lambda$ being strictly larger than zero is the presence of finite size fluctuations. We trace back the origin of the logarithmic correction to a weak synchronization between tangent and phase space dynamics.