Classification
The CML system evolves through discrete time by a mapping on vector sequences. These mappings are a recursive function of two competing terms: an individual non-linear reaction, and a spatial interaction (coupling) of variable intensity. CMLs can be classified by the strength of this coupling parameter(s).
Much of the current published work in CMLs is based in weak coupled systems where diffeomorphisms of the state space close to identity are studied. Weak coupling with monotonic (bistable) dynamical regimes demonstrate spatial chaos phenomena and are popular in neural models. Weak coupling unimodal maps are characterized by their stable periodic points and are used by gene regulatory network models. Space-time chaotic phenomena can be demonstrated from chaotic mappings subject to weak coupling coefficients and are popular in phase transition phenomena models.
Intermediate and strong coupling interactions are less prolific areas of study. Intermediate interactions are studied with respect to fronts and traveling waves, riddled basins, riddled bifurcations, clusters and non-unique phases. Strong coupling interactions are most well known to model synchronization effects of dynamic spatial systems such as the Kuramoto model.
These classifications do not reflect the local or global (GMLs ) coupling nature of the interaction. Nor do they consider the frequency of the coupling which can exist as a degree of freedom in the system. Finally, they do not distinguish between sizes of the underlying space or boundary conditions.
Surprisingly the dynamics of CMLs have little to do with the local maps that constitute their elementary components. With each model a rigorous mathematical investigation is needed to identify a chaotic state (beyond visual interpretation). Rigorous proofs have been performed to this effect. By example: the existence of space-time chaos in weak space interactions of one-dimensional maps with strong statistical properties was proven by Bunimovich and Sinai in 1988. Similar proofs exist for weakly hyperbolic maps under the same conditions.
Read more about this topic: Coupled Map Lattice
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