Normal Forms and Unfoldings for Local Dynamical Systems

In this chapter, two examples, one semisimple and the other not, will be treated from an elementary point of view. The purpose of the treatment is to motivate the concerns and themes of the remainder of the book. The semisimple example is the nonlinear center; when unfolded, this becomes the Hopf bifurcation. The nonsemisimple example is the generic double-zero eigenvalue, which unfolds to the Takens-Bogdanov bifurcation.

The examples in Chapter 1 have made it clear that it is important to be able to find a complement C to the image of a linear map L: V → V, that is, a space C such that This is purely a problem in linear algebra, and so can be discussed without any explicit mention of normal forms.

Consider a smooth (real or complex) matrix-valued function A(ε) of a (real) small parameter ε, having formal power series How do the eigenvectors (or generalized eigenvectors) and eigenvalues of such a matrix vary with ε? This question arises, for instance, in studying the stability of the linear system of differential equations ẋ = A(ε)x. Or the nonlinear system ẋ = f(x, ε) may have a rest point x*(e) whose stability depends on the matrix A(ε) = fx(x*(ε), ε). The same question arises for diíferent reasons in quantum mechanics; in this case A(ε) is Hermitian, hence diagonalizable, and the interest focuses on the splitting (for ε ≠ 0) of eigenvalues that are equal when ε = 0. (These eigenvalues can be, for example, the spectral lines of an atom, which can split in the presence of an external field.)

Chapters 2 and 3 have been somewhat of a digression from the problem posed in Chapter 1, that of normalizing a system of nonlinear differential equations. In this chapter we return to that problem, equipped with the methods discussed in Chapters 2 and 3.

After a system has been placed into normal form, an immediate question arises: What does the normal form tell us about the dynamics of the system? Sometimes the normalized system is simple enough to be integrable (“by quadrature,” that is, its solution can be reduced to the evaluation of integrals), but this is not the usual case (except in two dimensions). We approach this question in Section 5.1 by establishing the existence of geometrical structures, such as invariant manifolds and preserved foliations, for systems in truncated normal form (that is, polynomial vector fields that are entirely in normal form, with no nonnormalized terms). These geometrical structures explain why some truncated normal forms are integrable, and give partial information about the behavior of others. In particular, the computation of a normal form up to degree k simultaneously computes the stable, unstable, and center manifolds, the center manifold reduction of the system, the stable and unstable fibrations over the center manifold, and various preserved foliations. These concepts will be defined (to the extent that they are needed) when they arise, but the reader is expected to have some familiarity with them. We do not prove the existence of these structures, except in the special case of truncated systems in normal form, where each structure mentioned above takes a simple linear form. For the full (untruncated) systems the relevant existence theorems will be stated without proof, and the normal form will be used to compute approximations.

As the title indicates, this short final chapter is not intended to be a complete treatment of bifurcation theory, or even a complete overview. Instead, we focus on certain specific topics, chosen because they are closely related to the main themes of this book, or because they are not developed extensively in other monographs. Among these topics are the questions of jet sufficiency, the computation of unfoldings, and the rescaling of unfolded systems.