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apply approach zero asymptotically stable Banach space bounded Chapter characteristic exponents characteristic multipliers co-limit set coefficients compact set completes the proof continuous first derivatives continuous function Corollary defined derivatives with respect determine differential equations discuss eigenvalues equilibrium point equivalent Exercise existence fact fixed point follows fundamental matrix fundamental matrix solution Furthermore given homeomorphism hypothesis initial value integral manifold interval Lemma Liapunov function linear system linear variational equation lipschitzian mapping negative real neighborhood noncritical with respect nonlinear nonsingular obtain open set partial derivatives periodic orbit periodic solution perturbation positive definite principal matrix solution problem proof of Theorem proves the lemma R X Rk real numbers saddle point scalar Section sequence solution of 1.1 solution of period solution x(t stability properties subset Suppose T-periodic Theorem 2.1 theory tions torus transformation uniformly asymptotically stable unique solution unstable manifolds w-vector x(to