In this paper it is proved that if a one-parameter family (F1) of C1dissipative maps in dimension two creates a new homoclinic intersection for a fixed point Ptwhen the parameter t = t0, then there is a cascade of quasi-sinks, i.e., there are parameter values tn converging to t0such that, for t = tn, Ft has a quasi-sink An with each point q in An having period ti. A quasi-sink Anfor a map F is a closed set such that each point q in Anis a periodic point and Anis a quasi-attracting set (å la Conley), i.e., Anis the intersection of attracting sets Formula Present where each Formula Present has a neighborhood Formula Present. Thus, the quasi-sinks An are almost attracting sets made up entirely of points of period n. Gavrilov and Silnikov, and later Newhouse, proved this result when the new homoclinic intersection is created nondegenerately. In this case the sets Anare single, isolated (differential) sinks. In an earlier paper we proved the degenerate case when the homoclinic intersections are of finite order tangency (or the family is real analytic), again getting a cascade of sinks, not just quasi-sinks.
ASJC Scopus subject areas
- Applied Mathematics