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Discretize and solve Chimera state model

  1. Oct 30, 2008 #1
    Hi all,

    We are working on a paper - Chimera States for Coupled Osscillators from Daniel M. Abrams and Steven H. Strogatz.
    We have the following formula of the simplest system that supports a chimera state:
    [tex]\frac{\partial\phi}{\partial t} = \omega - \displaystyle\int^\pi_{-\pi} G(x-x')\sin[\phi(x,t)-\phi(x',t)+\alpha]dx'[/tex]
    Now we have to discretize it and 'ODE' it.
    Our discretization looks like:
    [tex]\dot{\phi_i} = \omega_i - \frac{1}{N}\displaystyle\sum_{j=1}^N G(x_i-x_j')\sin[(\phi(x_i,t)-\phi(x_j',t)+\alpha][/tex]
    Where [tex]\phi[/tex] is the phase of the oscilator at position x at time t, i is the current oscillator and j are the oscillators in the ring.
    [tex]G(x) = \frac{1}{2\pi}(1+A\cos x)[/tex]
    Now we have to solve this equation with [tex]x = \pm \pi \quad \alpha = \frac{\pi}{2} - 0.18 \quad A = 0.995 \quad N=256[/tex]
    The solver Runge-Kutta has to be used with dt = 0.025 for 200000 iterations and starting from [tex]\phi(x)=6rexp(-0.76x^2)[/tex]. r is a random variable between -0.5 and 0.5

    Who can help us.

  2. jcsd
  3. Jan 13, 2009 #2
    Try downloading the free software XPPAUT and solve using it.
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