Network of globally coupled maps
Research and application of principled of information processing in natural neural networks (brains of humans and animals) is one of fundamental directions of science. One approach may be based on the careful reproduction of details revealed in biological systems. Alternatively, one can try to adopt from biology only the most general points like existence of a huge number of cells (neurons) and of a system of coupling between them. Of a definite interest from this point of view is the model suggested by Kaneko. Each individual cell is represented by a logistic map that can demonstrate complex dynamics and chaos, while the coupling is global: each cell is connected in the same way with each other. This may be thought as well as presence of a common mean field effecting in the same manner all the elements of the system.
One advantage of the globally coupled network is a simplicity of its practical design. For example, in electronic systems the global coupling may be arranged easily through the common power supply circuit, or by placing the whole system into the common resonator.
Model with two types of global coupling
In the original work of Kaneko and in many subsequent studies the dissipative type of coupling was postulated. However, as follows from our analysis of two coupled maps it is worth considering a model including two types of coupling, inertial and dissipative. Let us introduce a network of globally coupled maps by equations:
where f(x)=1-lx2 is nonlinear function associated with the logistic map n designates discrete time, index i numerates the cells of the network, N is the whole number of the cells, e1 and e2 are coupling parameters. Two last terms (sums) in the equation do not depend on the index, in other words, they are the same for all the cells. So, they may be interpreted as two mean fields associated with the two types of global coupling:
Clusterization
In the model with dissipative global coupling Kaneko has discovered a phenomenon called clusteriyation. It consists in spontaneous formation of groups of cells - clusters in such way that instantaneous states of cells relating to the same cluster coincide exactly. It is possible due to the global nature of coupling because neither the states of the cells, nor the effecting mean field differ for the cells in the same cluster. The same phenomenon obviously exists also in the model with two types of coupling because the cells which have coinciding instantaneous states are in identical conditions independently of the number of mean fields.
It is natural to classify possible regimes of the system by a number of clusters
K and by relative populations of the clusters (this is the ratio of a number of cells relating to the given cluster to the whole number of the cells) pm=nm/n. Description of dynamics in the case of small number of clusters K can be simplified significantly, because the sums over the clusters are estimated trivially. Thus, we arrive at a set of K coupled maps. 
where k=1,2,...k, Xnk relates to the k-th cluster, and sum of pk equals 1. The simplest example is a two-cluster state with relative populations p1 and p2=1p1:
In particular, for p1=p2=1/2 this is the system discussed in the section devoted to two coupled maps.
Kaneko phases
For characterization of regimes arisen in dependence on parameters in the globally coupled network let us turn to the concept of phases introduced by Kaneko. For a given point in the parameter space (l, e1, e2) we consider an ensemble of identical mutually independent globally coupled networks with random initial conditions. After a sufficiently large number of iterations we estimate statistics of the number of clusters.
System with global dissipative coupling:
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System with global inertial coupling: 
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The phase diagrams shown here are obtained in computer experiment, the phases are designated by colors and marked by appropriate letters. These are cross-sections of the complete parameter space by a plane (l, e1=0, e2=ed) in the first picture (dissipative coupling), and by a plane (l, e1=ei, e2=0.088ei) in the second one (inertial coupling).
Scaling
As the globally coupled network may be regarded as a set of cells connected pairwise, the scaling regularities should be valid for these systems too, like in two coupled maps or in coupled map lattices.
Let us suppose that at the parameter values
l,
eI,
eD we detect some Kaneko
phase for an ensemble of systems with random initial conditions from the
interval
|x|<C.
Then, for an ensemble with random initial conditions in the interval
|x|<C/|a|
at the point
The right-side diagrams in the both figures represent the results of computations with rescaling of the parameters and initial conditions in accordance with the formulated rules. Comparing the pictures disposed side by side you can see how the scaling property works.
Saratov group
