Origin and scaling of chaos in weakly coupled phase oscillators.

Abstract

We discuss the behavior of the largest Lyapunov exponent λ 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 λ∼1/N, while for strong fluctuations of the frequency spacing λ∼lnN/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 λ 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.