Heterogeneous, Weakly Coupled Map Lattices

TL;DR

This paper investigates heterogeneous weakly coupled map lattices, demonstrating that their periodic orbits remain stable under small coupling, even amidst complex bifurcation phenomena, thereby advancing understanding of their bifurcation structures.

Contribution

It characterizes periodic orbits and bifurcation behavior in heterogeneous weakly coupled map lattices, relaxing the assumption of identical component maps.

Findings

Periodic cycle periods are preserved under small coupling.

Periodic orbits are characterized near and far from saddle-node bifurcations.

Provides insights into the bifurcation structure of heterogeneous CMLs.

Abstract

Coupled map lattices (CMLs) are often used to study emergent phenomena in nature. It is typically assumed (unrealistically) that each component is described by the same map, and it is important to relax this assumption. In this paper, we characterize periodic orbits and the laminar regime of type-I intermittency in heterogeneous weakly coupled map lattices (HWCMLs). We show that the period of a cycle in an HWCML is preserved for arbitrarily small coupling strengths even when an associated uncoupled oscillator would experience a period-doubling cascade. Our results characterize periodic orbits both near and far from saddle--node bifurcations, and we thereby provide a key step for examining the bifurcation structure of heterogeneous CMLs.