Homoclinic bifurcations in n-dimensional endomorphisms, due to expanding periodic points

Abstract

The study of bifurcation mechanisms which determine the transition from regular to chaotic behaviour (or “route to chaos”) in endomorphisms (maps with a nonunique inverse) has reached a good level of theoretical knowledge for one-dimensional maps. We refer to the book by Mira [l] as a review of the main results in this research area. The situation is quite different as regards endomorphisms of dimensions higher than one. A few results have been worked out until now [2-81, mainly in the simplest cases of two-dimensional endomorphisms. However, many mathematical models of interest in the applications are ultimately described by maps with a nonunique inverse. Clearly, in the plane IR2, or in higher-dimensional spaces, we can no longer take advantage of the Koenigs-Lemeray graphical construction, and of the qualitative behaviour of the graphs of powers of t(x) (t’(x), t3(x), . ..). which are powerful tools to identify the local and global bifurcations of one-dimensional endomorphisms, x’ = t(x), x E IR’. However, to study bifurcation mechanisms in IR”, n 2 2, we make use of properties which are the natural generalizations of those used in the one-dimensional case. Indeed, a basic tool for studying the dynamics and bifurcation mechanisms of endomorphisms is provided by (the analysis of) the critical manifolds, first introduced by Gumowski and Mira in 1965 (see [2] and references therein). The critical manifolds are the natural generalization to IR”, n 2 2, of the local extrema (also called critical points of rank-l) of one-dimensional endomorphisms. In this work, we deal with homoclinic bifurcations in endomorphisms of IR” (n 1 l), due to expanding periodic points. As in one-dimensional endomorphisms, we will see that critical manifolds play a pre-eminent role. Our purpose is to contribute to the understanding of the bifurcation mechanisms related to homoclinic orbits occurring in n-dimensional endomorphisms, mechanisms strictly related to the nonuniqueness of the inverse function, that is, typical of endomorphisms, not present in invertible maps, although similar to those occurring in these maps. In particular, this work concerns the global bifurcation (homoclinic bifurcations) occurring when a repelling fixed point becomes a “snap-back repeller”t (SBR henceforth), and in general when there exists a critical orbit homoclinic to an expanding