Nonlinear Systems: Bifurcation Theory

Abstract

In Chapters 2 and 3 we studied the local and global theory of nonlinear systems of differential equations (1) with f∈C1 (E) where E is an open subset of R n . In this chapter we address the question of how the qualitative behavior of (1) changes as we change the function or vector field f in (1). If the qualitative behavior remains the same for all nearby vector fields, then the system (1) or the vector field f is said to be structurally stable. The idea of structural stability originated with An- dronov and Pontryagin in 1937. Their work on planar systems culminated in Peixoto’s Theorem which completely characterizes the structurally stable vector fields on a compact, two-dimensional manifold and establishes that they are generic. Unfortunately, no such complete results are available in higher dimensions (n ≥3). If a vector field f∈C 1(E) is not structurally stable, it belongs to the bifurcation set in C l(E). The qualitative structure of the solution set or of the global phase portrait of (1) changes as the vector field f passes through a point in the bifurcation set. In this chapter, we study various types of bifurcations that occur in C 1-systems (2)depending on a parameter µ∈R (or on several parameters µ∈R m ). In particular, we study bifurcations at nonhyperbolic equilibrium points and periodic orbits including bifurcations of periodic orbits from nonhyperbolic equilibrium points. These types of bifurcations are called local bifurcations since we focus on changes that take place near the equilibrium point or periodic orbit. We also consider global bifurcations in this chapter such as homoclinic loop bifurcations and bifurcations of limit cycles from a one- parameter family of periodic orbits such as those surrounding a center. This chapter is intended as an introduction to bifurcation theory and some of the simpler types of bifurcations that can occur in systems of the form (2). For the more general theory of bifurcations, the reader should consult Guckenheimer and Holmes [G/H],Wiggins [Wi], Chow and Hale [C/H], Golubitsky and Guillemin [G/G] and Ruelle [Ru].