On Bifurcations of Two - Dimensional Di eomorphismswith a Homoclinic

|Bifurcations of periodic orbits are studied for two-dimensional diieomorphisms close to a diieomorphism with the quadratic homoclinic tangency to a saddle xed point whose Jacobian is equal to one. Problems of the coexistence of periodic orbits of various types of stability are considered. INTRODUCTION It is well known that homoclinic bifurcations of two-dimensional diieomorphisms generally give rise to either stable or completely unstable periodic orbits 1]. In this case, the type of stability depends on whether the saddle quantity (the modulus of the product of multipliers) of a saddle xed (periodic) point with a homoclinic tangency is greater or less than one. It is also known that, when < 1, close diieomorphisms have no completely unstable (or stable when > 1) periodic orbits in a small neighborhood of a nontransversal homoclinic orbit. In the case of a saddle xed point of neutral type, i.e., when = 1 at the moment of tangency, the main bifurcations within two-parameter families were investigated in our paper 2 2]. In this study, we showed that a variation of parameters in this case may lead not only to stable and completely unstable orbits but also to invariant closed curves. In the present paper, these studies are continued. The main attention is paid to the analysis of basic elements of the structure of bifurcation diagrams and related problems of the coexistence of periodic orbits for various types of stability. The contents of this paper are as follows. Section 1 contains the statement of the problem and a short review of the results obtained in 2]. Section 2 describes the bifurcations of xed points of rst return maps. Section 3 deals with the structure of bifurcation diagrams for single-round periodic orbits.