Coupled Map Lattice and Globally Coupled Map

After downloading maplattice.jar, please execute it by double-clicking, or typing "java -jar maplattice.jar".

If the above application does not start, please install OpenJDK from

CML (coupled map lattice) and GCM (globally coupled map) are the discrete-time map written by

respectively. CML couples the N chaotic maps xn+1=f(xn) locally, and GCM couples them globally.

As the chaotic map f(x), let us consider the logistic map f(x) = 1-a x2. Note that this map is equivalent to the well-known form g(z) = r z(1-z)
when the transformations a=r(r-2)/4 and x=2(2z-1)/(r-2) are taken.

These models were proposed by Dr. Kunihiko Kaneko, to use as general models for the complex high-dimensional dynamics, such as biological systems, networks of DNA, economic activities, neural networks, evolution, and so on.

The nonlinear dynamics and the chaos theory succeeded in understanding the low-dimensional complex dynamics where the number of variables which govern the system (e.g, 3 in Lorenz equation), or the number of positive Lyapunov exponents (e.g, 1 in Lorenz attractor), is small.
In such systems, the knowledge based on the analyses of the low-dimensional universal systems, e.g., logistic map, was efficiently utilized, but, it is insufficient for the analysis of the high-dimensional complex system, and a new recipe would be required.
The riddled basin, the on-off intermittency, the coupled map lattice (CML), and the globally coupled map (GCM) would be examples of them.

CML and GCM connect the many chaotic maps. To judge whether such a violent simplification is effective in modeling the nature, we should wait for the progress of the research on complex system. It is important that the chaotic itinerancy, which is often observed in many high-dimensional chaotic systems, was found in GCM.

This simulator shows the behavior of the model with 200-coupled logistic maps.
You can select CML or GCM with the upper choice, and you can change the values of the nonlinear parameter a and the coupling strength g with the right bars.
The output xn(i) of the site i is shown in every two time steps. The blue color means the large xn(i).

The following phenomena are typically observed.


When the coupling strength g is increased, the transition is observed. With the parameters with medium values, chaotic itinerancy is also observed (e.g., a=1.95, g=0.2).
This is a phenomenon where the synchronized clusters are sometimes rearranged chaotically, which is often observed in the high-dimensional dynamical system where many chaotic attractors co-exist. The chaotic pattern transitions in associative memory would be an example of the chaotic itinerancy.

In both CML and GCM, the many attractors co-exist in the high-dimensional phase space.

This page is based on the following textbook.
<< Blowout Bifurcation and On-off Intermittency / Double Pendulum >>

Introduction to Chaos and Nonlinear Dynamics