# Discretize and solve Chimera state model

1. Oct 30, 2008

### Chaos08

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:
$$\frac{\partial\phi}{\partial t} = \omega - \displaystyle\int^\pi_{-\pi} G(x-x')\sin[\phi(x,t)-\phi(x',t)+\alpha]dx'$$
Now we have to discretize it and 'ODE' it.
Our discretization looks like:
$$\dot{\phi_i} = \omega_i - \frac{1}{N}\displaystyle\sum_{j=1}^N G(x_i-x_j')\sin[(\phi(x_i,t)-\phi(x_j',t)+\alpha]$$
Where $$\phi$$ 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.
$$G(x) = \frac{1}{2\pi}(1+A\cos x)$$
Now we have to solve this equation with $$x = \pm \pi \quad \alpha = \frac{\pi}{2} - 0.18 \quad A = 0.995 \quad N=256$$
The solver Runge-Kutta has to be used with dt = 0.025 for 200000 iterations and starting from $$\phi(x)=6rexp(-0.76x^2)$$. r is a random variable between -0.5 and 0.5

Who can help us.

Regards

2. Jan 13, 2009

### synchro

Try downloading the free software XPPAUT and solve using it.