An existence theorem of smooth nonlocal center manifolds for systems close to a system with a homoclinic loop (Q1306503)

Considered are systems of differential equations close to a system that has a homoclinic loop. The study of the dynamics of such systems is facilitated by a dimensional reduction achieved through the construction of a locally invariant center manifold that contains all limit sets. Such invariant center manifolds were constructed by a number of people: The theses of Turaev and Sandstede, the reviewer [Mem. Am. Math. Soc. 578 (1996; Zbl 0862.34042)] and the paper [J. Differ. Equ. 126, 1-47 (1996; Zbl 0849.34036)] by \textit{R. Rousseau} and \textit{C. Roussarie}. In this paper the authors give a detailed proof of the existence of invariant center manifolds for systems of differential equations that are merely continuously differentiable. Orbits are controled by considering them as solutions to boundary value problems, a successful approach in bifurcation theory developed in this context by Shil'nikov.