A coupled
map lattice (CML) is a
dynamical system that models the behavior of
non-linear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
systems (especially
partial differential equations). They are predominantly used to qualitatively study the
chaotic dynamics
Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kids, ...
of spatially extended systems. This includes the dynamics of
spatiotemporal
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
chaos where the number of effective
degrees of freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
diverges as the size of the system increases.
Features of the CML are
discrete time dynamics, discrete underlying spaces (lattices or networks), and real (number or vector), local, continuous
state variables. Studied systems include
populations,
chemical reactions,
convection,
fluid flow
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
and
biological networks. More recently, CMLs have 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 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
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
. However, the similarities can be compounded when considering multi-component dynamical systems.
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 map
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
ping 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 parameter r is slightly increased with each time step. The result is a raw form of chaotic behavior in a map lattice. 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 computed from the recursive maps both on
itself and on the neighboring site
. The coupling parameter
is equally weighted. Again, the value of
is constant 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).
History
CMLs were first introduced in the mid 1980s through a series of closely released publications. Kapral used CMLs for modeling chemical spatial phenomena. Kuznetsov sought to apply CMLs to electrical circuitry by developing a
renormalization group approach (similar to Feigenbaum's
universality to spatially extended systems). Kaneko's focus was more broad and he is still known as the most active researcher in this area. The most examined CML model was introduced by Kaneko in 1983 where the recurrence equation is as follows:
: