Couple Map Lattices

by Chris Ormerod, Bernard Pailthorpe and Nicole Bordes

Introduction
One of the simplest non-linear systems is described by the logistic equation

also called a 'map'. An iteration is defined as x-> f(x), and over a number of iterations we can see that chaotic nature of the system. The system may settle into certain values called stable points, or the system may cycle through a finite sequence of values in which we call a cycle, and these stable points and cycles are dependant on one variable, alpha, which is called the logistic constant.

A mapping of the values of the stable points and the cycles that this system should settle into for a given alpha is the well known Bifrication Diagram.

The system
The following document is concerned with the discoveries made concerning chaotic systems defined on coupled lattice maps. The system used was basically a 2 dimensional descrete valued latice in which every square on the lattice takes a value from zero to one where the function applied is defined as follows.
    - f[x] denotes the logistic function ie f[x]=cx(1-x)
    - Each value in the square on the descrete map is x(i,j) where i denotes the i-th row, and j denotes the j-th column.
    - L is the set of all x(i,j), in our case i,j is on the closed interval from 0 to 19.
    - F[L,c,d] is defined as, for every value in L,
            x(i,j)->(1-4d)f[x(i,j)] + f[x(i+1,j)]+f[x(i,j+1)]+f[x(i-1,j)]+ f[x(i,j-1)]
        for i,j defined mod 20. Thus the mapping is on a topological torus.
    - Each Transform is dependent on d, the coupling constant, and c, the logistic constant.
This implies the value of a single sqaure on this grid after one iteration coupled with the squares adjacent to it and the strength of the interactions between adjacent squares is constant.

Characterising the system
If we were to consider any properties of the following mapping, first we need a way to assign a characteristic "value" to a lattice . The value that we decided to assign to a given Latice L, is based on it's matrix representation. One of the simplest characteristics of a martix is its Trace, that is the sum of it's eigenvalues which is also the sum of the diagonal elements of the matrix. This was based on the assumption of the steady states that could be achieved. One of the steady states is as follows.
- All x(i,j) are equal to the corresponding value for a logistic cycle with constant c. In which the coupling constant does not affect the steady state. eg. for 2 cycle all values are equal to solution of this eqation.

    - The cycles interchange values periodically, but remain in some cycle defined by both the logistic and coupling constant. This is a little more difficult, it assumes that the steady state will eventually be in a set form. eg .
          a b a b
          b a b a A two cycle where x(i,j)=a,b
          a b a b depending on time t.

This implies a and b are the solutions to the two equations,

for given alpha and beta.

The solutions to the second set of equations seemed to describe the actual results to some extent, in our case the sum of the eigenvalues oscilate between 20*a and 20*b. When looking at the trace of the given matrix over time it was found that eventially the system would settle into a cycle and this occured even in randomly generated initial conditions. And further more the results agreed with values predicted in the system with interchanging values in adjacent squares.

This is an example of the cycle in T formed by the coupling constant of 0.01 and a logistic constant of 3.3 started from a random 9 by 9 map over 50 iterations, and as you can see the values settle into a steady state. But because the sum of the eigen values on this matrix is just the sum of the diagonal elements and the fact that the values are doubly periodic, the values obtained are found along all diagonals.

If we isolate these values for a certain value of the coupling constant and for variable logistic constant we obtain what looks like a Bifrication Diagram. These are the experimental results of this for the coupling constant equal to 0.1

This diagram reveals the presence of cycles greater than two and also show the existence of a chaotic region for some conditions. Although the presence of cycles more than two exist, an algerbraic solution to this equation was unattainable in the given time. An algebraic solution to the two cycle and stable points were obtained and plotted, this was in agreement with our experiemental results.

The behavior of these values seemed to vary over the both the coupling constant and the logistic constant. The resultant diagram could not be represented in 2-dimensions. The behavior in the coupling constant gave the graph a new dimension. So the object that describes the behavior of this chaotic system looks a little like this.

And this is an image from a different view point.

What this represents is the cycle of the Trace values in the Z-axis and the coupling constant and the logistic constant along the X and Y axes. This shows a complex behavior for large values of the logistic constant and low values for the coupling constant. In fact for high values of the coupling constant the behavior of this system is fairly regular.

A VRML 2 model is available here

Publication
This work was presented at HPC Asia 2001, Gold Coast, Sept 24-28 2001.

Chris Ormerod, N. Bordes, B. Pailthorpe 'Characterising Coupled Map Lattices' Proceedings of HPC Asia'01, Gold Coast, Australia (Sept. 24-28, 2001).