Heterogeneous, weakly coupled map lattices

• Many natural patterns arise through heterogeneous interactions between discrete elements.
• Coupled map lattices (CMLs) model interactions between discrete dynamical systems.
• We find analytical solutions to help describe the dynamics of heterogeneous CMLs.
• We examine synchronization in CMLs that undergo period-doubling cascades.
• We examine the laminar regime in Type-I intermittency in heterogeneous CMLs.

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.