## Abstract

In this paper it is proved that if a one-parameter family (F_{1}) of C^{1}dissipative maps in dimension two creates a new homoclinic intersection for a fixed point P_{t}when the parameter t = t_{0}, then there is a cascade of quasi-sinks, i.e., there are parameter values tn converging to t_{0}such that, for t = t_{n}, Ft has a quasi-sink An with each point q in An having period ti. A quasi-sink A_{n}for a map F is a closed set such that each point q in A_{n}is a periodic point and A_{n}is a quasi-attracting set (å la Conley), i.e., A_{n}is 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 A_{n}are 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.

Original language | English (US) |
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Pages (from-to) | 841-849 |

Number of pages | 9 |

Journal | Transactions of the American Mathematical Society |

Volume | 288 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1985 |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics