Stability of Homogeneous Solutions

Let us begin by considering the stability of homogeneous solutions and/or initial-conditions i.e. by considering the stability of a simple-diffusive CML when cri(O) = a for all sites i , where a is a fixed point of the local logistic map F(cr) = acr(l—cr). Following Waller and Kapral [kapral84], we first recast equations 8.23 and 8.24 

Using the orthogonality relation exp(27rzj(m — s)/N) = N6ms, where 6  

Now let us consider the stability of the two systems around fixed points of /(cr), and therefore around homogeneous solutions of the CML. From chapter 4 we recall that (7 = 0 is a stable fixed point for a 1 and cr = 1 - 1/a is a stable fixed point for 1 a 3. Let us see whether our diffusive coupling leads to any instability. 

Substituting equation 8.29 into equation 8.27 and retaining only those terms that are linear in Sr/mf t) we get 

We see from both equations 8.32 and 8.33 that the most unstable mode is the mode and that ai t) = 1 - 1/a is stable for 1 a 3 and ai t) = 0 is stable for 0 a 1. In other words, the diffusive coupling does not introduce any instability into the homogeneous system. The only instabilities present are those already present in the uncoupled local dynamics. A similar conclusion would be reached if we were to carry out the same analysis for period p solutions. The conclusion is that if the uncoupled sites are stable, so are the homogeneous states of the CML. Now what about inhomogeneous states  

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