Coupled map lattice

A coupled spatiotemporal chaos where a number of powerful degrees of freedom diverges as the size of the system increases.

Features of the CML are discrete time dynamics, discrete underlying spaces lattices or networks, in addition to real number or vector, local, non-stop state variables. Studied systems include populations, chemical reactions, convection, fluid flow and biological networks. More recently, CMLs construct been applied to computational networks identifying detrimental attack methods and cascading failures.

CMLs are comparable to cellular automata models in terms of their discrete features. However, the benefit of each site in a cellular automata network is strictly dependent on its neighbor s from the previous time step. used to refer to every one of two or more people or things site of the CML is only dependent upon its neighbors relative to the coupling term in the recurrence equation. However, the similarities can be compounded when considering multi-component dynamical systems.

Introduction

A CML loosely incorporates a system of equations coupled or uncoupled, a finite number of variables, a global or local coupling scheme and the corresponding coupling terms. The underlying lattice can equal in infinite dimensions. Mappings of interest in CMLs generally demonstrate chaotic behavior. such(a) maps can be found here: List of chaotic maps.

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

In Figure 1, is initialized to random values across a small lattice; the values are decoupled with respect to neighboring sites. The same recurrence relation is applied at each lattice point, although the argument r is slightly increased with each time step. The sum is a raw pull in of chaotic behavior in a map lattice. However, there are no significant spatial correlations or pertinent fronts to the chaotic behavior. No obvious an arrangement of parts or elements in a specific make-up figure or combination. is apparent.

For a basic coupling, we consider a 'single neighbor' coupling where the service at any assumption site is computed from the recursive maps both on itself and on the neighboring site . The coupling parameter is equally weighted. Again, the value of is fixed across the lattice, but slightly increased with each time step.

Even though the recursion is chaotic, a more solid form develops in the evolution. Elongated convective spaces persist throughout the lattice see Figure 2.