Representation by the Floquet Eigenvectors

Equation (3.3.5) represents a nonlinear phase diffusion equation. It is equivalent to the Burgers equation in the case of one space dimension (Chap. 6). It is known that the Burgers equation can be reduced to a linear diffusion equation through a transformation called the Hopf-Cole transformation (Burgers, 1974), and essentially the same is true for (3.3.5) in an arbitrary dimension. We shall take advantage of this fact in Chap. 6 when analytically discussing a certain form of chemical waves. 

The notion of asymptotic phase or isochrons has a certain relationship to the eigenvectors, defined for the linearized system of (2.1.1) about its periodic solution Xo(t). Consequently, various quantities which appear in our perturbation results, e.g., a and p, may be reinterpreted in terms of those eigenvectors. This new interpretation may sometimes prove convenient for analytical or numerical calculations based on a specific model (Sect. 3.5). Moreover, such a reformulation opens the way to a more systematic method of phase description as developed in Chap. 4. 

There is a theory called the Floquet theory (Cesari, 1971) which concerns first-order linear systems with periodic coefficients. In the present context, such a system arises from the linearization of (2.1.1) about. fo(f)- By putting X(t) = X() t)- u(t), this leads to 

Here S t) is a T-periodic matrix with the initial condition S(0) = 1, and A is some time-independent matrix. In the next chapter, the identity 

The eigenvalues are assumed to be algebraically simple, and one may require the orthonormality condition 

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