Onset of Oscillations in Distributed Systems

Then the solvability condition (2.2.16) itself turns out to take the form of the Stuart-Landau equation 

The foregoing argument was about systems of ordinary differential equations. For chemical reactions this corresponds to the dynamics of local systems which 

This is called a reaction-diffusion equation ) is a matrix of diffusion constants, and is often assumed to be diagonal. The specific problem we shall consider in the rest of this chapter concerns the possible roles of the many spatial degrees of freedom taken on when a uniform steady state experiences an instability of oscillatory type. In this section, we give only a qualitative argument the formal derivation of the Ginzburg-Landau equation is rather easy, and will be done in the next section. 

To make the point clear, we shall work in this section with the system (2.3.1) in an interval - /2 x /2, subject to the no-flux boundary conditions 

The stability of the uniform steady solution Xq of (2.3.1) may then be analyzed from the variational equations about 

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