Coupled Map Lattice - Introduction


A CML generally 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 exist in infinite dimensions. Mappings of interest in CMLs generally demonstrate chaotic behavior. Such 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 (see Logistic map). It is graphed across a small lattice and decoupled with respect to neighboring sites. The recurrence equation is homogeneous, albeit randomly seeded. The parameter r is updated every time step (see Figure 1, Enlarge, Summary):

The result is a raw form of chaotic behavior in a map lattice. The range of the function is bounded so similar contours through the lattice is expected. However, there are no significant spatial correlations or pertinent fronts to the chaotic behavior. No obvious order is apparent.

For a basic coupling, we consider a 'single neighbor' coupling where the value at any given site is mapped recursively with respect to itself and the neighboring site . The coupling parameter is equally weighted.

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

Figure 1: An uncoupled logistic map lattice
with random seeding over forty iterations.
Figure 2: A CML with a single-neighbor
coupling scheme taken over forty iterations.

Read more about this topic:  Coupled Map Lattice

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