Globally coupled chaotic maps and demographic stochasticity

TL;DR

This paper investigates how demographic stochasticity influences globally coupled chaotic maps, revealing a transition from period-doubling to slow dynamics as the number of agents increases, connecting stochastic effects with deterministic behavior.

Contribution

It introduces a two-step model analyzing demographic stochasticity in coupled chaotic maps, highlighting the emergence of slow dynamics and attractor degeneracy in the large population limit.

Findings

Small populations exhibit period-doubling cascades with increased coupling.

Large populations show extremely slow dynamics approaching deterministic solutions.

Degeneracy of attractors in deterministic systems arises from stochastic slow dynamics.

Abstract

The affect of demographic stochasticity of a system of globally coupled chaotic maps is considered. A two-step model is studied, where the intra-patch chaotic dynamics is followed by a migration step that coupled all patches; the equilibrium number of agents on each site, $N$, controls the strength of the discreteness-induced fluctuations. For small $N$ (large fluctuations) a period-doubling cascade appears as the coupling (migration) increases. As $N$ grows an extremely slow dynamic emerges, leading to a flow along a one-dimensional family of almost period 2 solutions. This manifold become a true solutions in the deterministic limit. The degeneracy between different attractors that characterizes the clustering phase of the deterministic system is thus the $N \to \infty$ limit of the slow dynamics manifold.