Spatial Chaos: Couple Lattice Maps
Spatial Chaos: Couple Lattice Maps
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.
Publication
Chris Ormerod, Bernard Pailthorpe, Nicole Bordes "Characterising coupled map lattices" HPC Asia Proceedings. Gold Coast, Sept 24-28 2001.
Credits:
- Chris Ormerod, Bernard Pailthorpe and Nicole
Bordes
Sydney VisLab, The University of Sydney
| 2001 |