Behaviour near a hyperbolic equilibrium

Abstract

In this chapter we start to investigate the behaviour of the nonlinear semiflow near an equilibrium. Throughout this chapter, and also in Chapters IX and X, we consider X over ℝ. For the linearized semiflow {T(t)}, the time asymptotic behaviour near the equilibrium u ≡ 0 is described by the decomposition of the state space according to the spectrum of the generator of the semigroup and the accompanying exponential dichotomy. In this chapter we will assume that there is no spectrum on the imaginary axis. In the case of RFDE, the spectrum in the right half-plane consists of finitely many, say k: eigenvalues (counting multiplicity) with a positive real part. From Chapter IV we recall that in this case we can decompose X as

$$X\, = {X_ - } \oplus {X_{ + \cdot }}$$