Differential Equations and Dynamical Systems

This chapter presents a study of linear systems of ordinary differential equations: where x ∈ R ⁿ, A is an n × n matrix and It is shown that the solution of the linear system (1) together with the initial condition x(0) = x₀ is given by where e ^At is an n × n matrix function defined by its Taylor series. A good portion of this chapter is concerned with the computation of the matrix e ^At in terms of the eigenvalues and eigenvectors of the square matrix A. Throughout this book all vectors will be written as column vectors and A ^T will denote the transpose of the matrix A.

In Chapter 1 we saw that any linear system has a unique solution through each point x_o in the phase space R ⁿ ; the solution is given by x(t) = e ^At x_o and it is defined for all t ∈ R.

In Chapter 2 we saw that any nonlinear system with ∈ C ¹(E) and E an open subset of R ⁿ , has a unique solution ø _t (x₀), passing through a point x₀ ∈ E at time t = 0 which is defined for all t ∈ I(x₀), the maximal interval of existence of the solution. Furthermore, the flow ø _t of the system satisfies (i) ø₀(x)=x and (ii) ø _t+s (x)=ø _t (ø _s (x)) for all x ∈ E; and the function ø(t, x)=ø _t (x) defines a C ¹-map ø: Ω→ E where Ω, = {(t, x) ∈ R × E|t∈I(x)}.

In Chapters 2 and 3 we studied the local and global theory of nonlinear systems of differential equations 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-systemsdepending 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].