Partial differential equations describing the time evolution of a physical system are very common. They are derived either directly from basic physical principles or as some reduced descriptions of microscopical evolutions valid in some limit. We shall not discuss the question of establishing these macroscopic equations, this is a very difficult problem which is not completely understood.

To fix the ideas, we mention some of the most important macroscopic equations with the idea that the theories discussed below will apply without too much changes to all these equations. One of the best known equations is the Navier-Stokes equation which describes...

We are, of course, interested in nontrivial solutions of the various equations mentioned in Chapter I. We will obtain various such solutions in the different stages of bifurcation theory developed in this book. Before we start with this program, we wish to prepare some material which will be needed essentially every time we talk about “small” solutions, or about the time evolution of small initial conditions. Intuitively, for small solutions, homogeneous nonlinearities (such asu³in the real amplitude equation) can be dropped and are “irrelevant” for the understanding of what is going on. The theory of invariant manifolds makes...

In this chapter, we review some of the standard material on bifurcation theory. The experienced reader may skip most of this material. Many good books on the subject exist, and we include the results mostly to fix terminology and notation. Some reference books are: [Ar1], [BPV], [GH], [IJ], and [R2].

We consider changes of a point attractor when external parameters vary. We shall first enlarge somewhat the discussion by analyzing the changes of stationary solutions of avector fieldwhich depends on parameters. A general answer is given by the following theorem.

Theorem 10.1.Let X(λ, ·)be a\mathcal {C}^{1}...

In Chapter III, we considered the time evolution of physical systems which have only a finite number of excited relevant degrees of freedom. The other degrees of freedom whose number may be infinite were, so to speak, slaves of the excited modes.

Now, we discuss systems in which an infinite number of degrees of freedom participate in the relevant physical evolution. In general, this occurs when we consider spatially infinitely extended systems. This is a situation which is reminiscent of Statistical Mechanics where one shows that large systems behave like infinite ones. Here, one can hope for a similar behavior,...

The next few sections deal with a very general outlook on the stability analysis of stationary (and by analogy of convective) solutions. We pursue here the following ideas.

a) A linear stability analysis can be done in great generality, using the method of Bloch functions, or time 2Π/ωmaps, when the stationary solution is periodic.

b) Because of invariances of the problem, the point 0 is seen to be in the spectrum of these excitations.

c) Depending on the bifurcation parameterɛand the spatial frequencyω, the spectrum near 0 may be stable or unstable. In the case of...

In Chapters I to V we have encountered a multitude of examples in which two phenomena were intimately connected:

Destabilization, and

Rescaling.

We want to argue now that this connection is not accidental, and we shall see in the next section and in Chapter VII an instance where 2 different scalings occur, leading to a so-called multiscale analysis.

The destabilization is always associated with a critical point in the underlying system. We have shown in particular that zero eigenvalues (or purely imaginary eigenvalues) are responsible for this destabilization. From the purely dimensional point of view, the rescaling becomes necessary because,...

If a physical system has several equilibrium states, or, as we have seen in earlier sections, several “quasistationary solutions,” then we can ask the following question: What happens in an (infinite) system which is one equilibrium state near to infinity on “one side” and in another state on the other side? There will be an “interface” between the two phases and we may then ask what the motion of this interface is. If, intuitively, one phase “wins” over the other by invading its region, then we shall say that a front propagates into that region. In general, a stable solution...