Homoclinic Tangencies, Ω-Moduli, and Bifurcations

|A survey of author's results related to the problems of existence of continuous invariants (moduli) of-conjugacy of multidimensional diieomorphisms with homoclinic tan-gencies is presented. The problem of bifurcations of periodic orbits is considered in the case of four-dimensional diieomorphisms with a nontransversal homoclinic orbit to a xed point of saddle{focus type. INTRODUCTION It is well known that bifurcations of multidimensional systems show a lot of surprise as compared with the classical bifurcations of two-dimensional ows. In particular, (1) structurally unstable multidimensional systems may form domains in the space of smooth dynamical systems (the so-called stable structural instability), and (2) structurally unstable multidimensional systems may admit moduli (continuous invariants) of topological and-equivalence. The latter fact is very important for the theory of bifurcations as a whole since the moduli are natural parameters that control bifurcations: any variation in the value of a modulus implies a variation in the structure of the set of orbits. This is especially important when there exist moduli of-equivalence since any variation in the value of the-modulus leads to bifurcations of nonwandering (periodic, homoclinic, etc.) orbits. An interest in the moduli of topological equivalence (conjugacy) arose in the late 1970s, after Palis 1] found out that such moduli of multidimensional systems appear every time the invariant manifolds of saddle periodic orbits intersect nontransversally. At that time (from 1970s to 1980s), systems with simple dynamics (without homoclinic orbits) were of primary interest. In particular, certain conditions were obtained under which such a system has either nite or innnite number of moduli (see, e.g., 2]). However, the-moduli were not actually considered, despite the fact that they were discovered earlier than the concept of modulus itself was introduced in the theory of dynamical systems. For instance, Gavrilov and Shil'nikov 3] showed that, in the class of systems