The continuation theory

Abstract

In the dissipative context, volume preserving context with p = 2 (hyperbolic case) or p > 2, Hamiltonian isotropic context, and reversible context 1, it is interesting to consider the restrictions of the unperturbed family of vector fields X and perturbed one $\tilde X$ to the center manifold. Here we have excluded the volume preserving context with p = 1 and p = 2 (elliptic case) since for these cases, the center manifold always coincides with the whole phase space. The center manifold persists under perturbations [67,115,158,356] but becomes, generally speaking, only finitely differentiable [12,347]. However, we can apply the finitely differentiable versions of the “relaxed” Theorems 2.8, 2.9, 2.11, 2.12 to the restrictions of X and $\tilde X$ to the center manifold, see [151, 243,277,278,306] as well as [62,162].