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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: : \qquad x_ = r x_n (1-x_n) In Figure 1, x_0 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 s is computed from the recursive maps both on s itself and on the neighboring site s-1 . The coupling parameter \epsilon = 0.5 is equally weighted. Again, the value of r is constant across the lattice, but slightly increased with each time step. : \qquad x_ = (\epsilon) x_n (1-x_n)s + (1-\epsilon) x_n (1-x_n) 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: : u_s^ = (1-\varepsilon)f(u_s^t)+\frac\left(f(u_^t)+f(u_^t) \right) \ \ \ t\in \mathbb,\ \varepsilon \in ,1/math> where u_s^t \in \ , and f is a real mapping. The applied CML strategy was as follows: * Choose a set of field variables on the lattice at a macroscopic level. The dimension (not limited by the CML system) should be chosen to correspond to the physical space being researched. * Decompose the process (underlying the phenomena) into independent components. * Replace each component by a nonlinear transformation of field variables on each lattice point and the coupling term on suitable, chosen neighbors. * Carry out each unit dynamics ("procedure") successively.


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 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 ...
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 coupled hyperbolic maps under the same conditions.


Unique CML qualitative classes

CMLs have revealed novel qualitative universality classes in (CML) phenomenology. Such classes include: * Spatial bifurcation and frozen chaos * Pattern Selection * Selection of zig-zag patterns and chaotic diffusion of defects * Spatio-temporal intermittency * Soliton turbulence * Global traveling waves generated by local phase slips * Spatial bifurcation to down-flow in open flow systems.


Visual phenomena

The unique qualitative classes listed above can be visualized. By applying the Kaneko 1983 model to the logistic {f(x_n)} = 1 - ax^2 map, several of the CML qualitative classes may be observed. These are demonstrated below, note the unique parameters: {, class="wikitable" border="1" , - , Frozen Chaos , Pattern Selection , Chaotic Brownian Motion of Defect , - , , , , - , Figure 1: Sites are divided into non-uniform clusters, where the divided patterns are regarded as attractors. Sensitivity to initial conditions exist relative to ''a'' < 1.5. , Figure 2: Near uniform sized clusters (''a'' = 1.71, ''ε'' = 0.4). , Figure 3: Defects exist in the system and fluctuate chaotically akin to Brownian motion (''a'' = 1.85, ''ε'' = 0.1). , - , Defect Turbulence , Spatiotemporal Intermittency I , Spatiotemporal Intermittency II , - , , , , - , Figure 4: Many defects are generated and turbulently collide (''a'' = 1.895, ''ε'' = 0.1). , Figure 5: Each site transits between a coherent state and chaotic state intermittently (''a'' = 1.75, ''ε'' = 0.6), Phase I. , Figure 6: The coherent state, Phase II. , - , Fully Developed Spatiotemporal Chaos , Traveling Wave , - , , , - , Figure 7: Most sites independently oscillate chaotically (''a'' = 2.00, ''ε'' = 0.3). , Figure 8: The wave of clusters travels at 'low' speeds (''a'' = 1.47, ''ε'' = 0.5).


Quantitative analysis quantifiers

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 * The power spectrum in space and time * Lyapunov spectra * Dimension density * Kolmogorov–Sinai entropy density * Distributions of patterns * Pattern entropy * Propagation speed of finite and infinitesimal disturbance * Mutual information and correlation in space-time *
Lyapunov exponent In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with ini ...
s, localization of
Lyapunov vector In applied mathematics and dynamical system theory, Lyapunov vectors, named after Aleksandr Lyapunov, describe characteristic expanding and contracting directions of a dynamical system. They have been used in predictability analysis and as initial ...
s * Comoving and sub-space time
Lyapunov exponent In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with ini ...
s. * Spatial and temporal
Lyapunov exponent In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with ini ...
s


See also

* Cellular automata *
Lyapunov exponent In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with ini ...
* Stochastic cellular automata *
Rulkov map The Rulkov map is a two-dimensional iterated map used to model a biological neuron. It was proposed by Nikolai F. Rulkov in 2001."Modelling of spiking-bursting neural behavior using two dimensional map/ref> The use of this map to study neural ne ...
*
Chialvo map The Chialvo map is a two-dimensional map proposed by Dante R. Chialvo in 1995 to describe the generic dynamics of excitable systems. The model is inspired by Kunihiko Kaneko's Coupled map lattice (CML) numerical approach which considers time ...


References


Further reading

* *
Alt URL
* *
Introduction to Chaos and Nonlinear Dynamics


External links


Kaneko Laboratory

Institut Henri Poincaré, Paris, June 21 – July 2, 2004

Istituto dei Sistemi Complessi
Florence, Italy
Java CML/GML web-app

AnT 4.669 – A simulation and Analysis Tool for Dynamical Systems
{{DEFAULTSORT:Coupled Map Lattice Nonlinear systems Chaotic maps