Extensive numerical investigations on the ergodic properties of two coupled Pomeau-Manneville maps

TL;DR

This paper numerically investigates the ergodic properties of two coupled Pomeau-Manneville maps, revealing anomalous statistics, dimensional scaling effects, and stretched exponential behaviors, with implications for understanding complex dynamical systems.

Contribution

It provides the first extensive numerical analysis of coupled Pomeau-Manneville maps, highlighting novel scaling laws and behaviors that differ from single intermittent maps.

Findings

Densities lack singularities at the marginal fixed point.

Escape and recurrence times follow power-law decay with dimensional scaling.

Lyapunov exponent convergence exhibits stretched exponential behavior.

Abstract

We present extensive numerical investigations on the ergodic properties of two identical Pomeau-Manneville maps interacting on the unit square through a diffusive linear coupling. The system exhibits anomalous statistics, as expected, but with strong deviations from the single intermittent map: Such differences are characterized by numerical experiments with densities which {\it do not} have singularities in the marginal fixed point, escape and Poincar\'e recurrence time statistics that share a power-law decay exponent modified by a clear {\it dimensional} scaling, while the rate of phase-space filling and the convergence of ensembles of Lyapunov exponents show a {\it stretched} instead of pure exponential behaviour. In spite of the lack of rigorous results about this system, the dependence on both the intermittency and the coupling parameters appears to be smooth, paving the way for further analytical development. We remark that dynamical exponents appear to be independent of the (nonzero) coupling strength.