Normally elliptic singular perturbations and persistence of homoclinic orbits
This paper deals with a dynamical system, possibly infinite dimensional or non-autonomous, with fast and slow time scales which is oscillatory with high frequencies in the fast directions. Firstly, the limit system of the slow variables is derived and justified. Further, there are constructed the stable, the unstable, the center-stable, the center-unstable and the center manifold of the steady state of a size of order \(O(1)\), and their leading order approximations are given. Finally, using these tools, persistence of homoclinic solutions in this type of normally eliptic singular perturbation problem is investigated. In an appendix, a process to block-diagonalize the linearized system of the singularly perturbed system under consideration at a steady state is outlined.
normally eliptic singular perturbation problems
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