       Coupled Map Lattice (Pls. Check the Mathematica Code and advice)

• To: mathgroup at smc.vnet.net
• Subject: [mg81339] Coupled Map Lattice (Pls. Check the Mathematica Code and advice)
• From: B^3 <bbbld at rediffmail.com>
• Date: Wed, 19 Sep 2007 05:40:25 -0400 (EDT)

```I am trying a code as per Mathematuca Guide for Graphics.. but not
sure wheter it will work or not.. even for low range/iterations it
takes tooo long time.................. (the problem is given at the
end).
Here goes the code:

Module[{rang =  500, k = 0.9, omega = 0.068, epsilon = 0.3,     incr = 0.2},

step[l_, epsilon_, f_] := (1 - epsilon)f(l) + (epsilon/2)
(f[RotateRight[l]] + f[RotateLeft[l]]);

f = FractionalPart[# - k/(2Pi)Sin[2Pi#] + omega] &;

lembda = Abs[Sin[7^(1/7)Range[rang]]];

tabul = Table[lembda = step[lembda, epsilon, f];
{#, j} & /@ Flatten[Position[Abs[Subtract @@@ Partition[lembda, 2,
1]], _?(# > incr &)]], {j,rang}];

Show[Graphics[Map[Rectangle[# - .5, # + .5] &, tabul, {2}]],
AspectRatio -> Automatic]]

The Problem:

We have a continuous variable x_i(t) at each site i at time t where
1<=i<=N. The evolution of x_i(t) is defined by

x_i(t+1) = F[x_i(t)] - (epsilon/2)[x_(i-1)(t) +x_(i+1)(t) - 2x_i(t)]

The parameter 'epsilon' is the coupling strength and the function
F(x)
is the circle map
F(x)= x + omega -(k/2*Pi)sin(2*Pi*x)

The dynamics is confined to the interval [0,1] using
If  int[x_i(t)]=m, x_i(t)=x_i(t)-m if x_i(t) >0

x_i(t)=x_i(t)-m+1 if x_i(t)<0

The fixed point solution of for the local map F(x) is given by

x* = (1/2*Pi) sin(-1)(2*Pi*omega/k)

My Problem is to draw "Space-Time" plot for the system, for say
omega=0.068, epsilon=0.3, k=0.9, N=500.

```

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