Coupled map lattice

They are predominantly used to qualitatively study the chaotic dynamics of spatially extended systems.

This includes the dynamics of spatiotemporal chaos where the number of effective degrees of freedom diverges as the size of the system increases.

[2] Studied systems include populations, chemical reactions, convection, fluid flow and biological networks.

More recently, CMLs have been applied to computational networks [3] identifying detrimental attack methods and cascading failures.

[4] However, the value of each site in a cellular automata network is strictly dependent on its neighbor(s) from the previous time step.

Each site of the CML is only dependent upon its neighbors relative to the coupling term in the recurrence equation.

A logistic mapping demonstrates chaotic behavior, easily identifiable in one dimension for parameter r > 3.57: In Figure 1,

The same recurrence relation is applied at each lattice point, although the parameter r is slightly increased with each time step.

Kuznetsov sought to apply CMLs to electrical circuitry by developing a renormalization group approach (similar to Feigenbaum's universality to spatially extended systems).

These mappings are a recursive function of two competing terms: an individual non-linear reaction, and a spatial interaction (coupling) of variable intensity.

Much of the current published work in CMLs is based in weak coupled systems [2] 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.

[10] Weak coupling unimodal maps are characterized by their stable periodic points and are used by gene regulatory network models.

These classifications do not reflect the local or global (GMLs [11]) coupling nature of the interaction.

With each model a rigorous mathematical investigation is needed to identify a chaotic state (beyond visual interpretation).

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.

[13] Similar proofs exist for weakly coupled hyperbolic maps under the same conditions.

These are demonstrated below, note the unique parameters: Coupled map lattices being a prototype of spatially extended systems easy to simulate have represented a benchmark for the definition and introduction of many indicators of spatio-temporal chaos, the most relevant ones are