TOWARDS GLOBAL-MODELS NEAR HOMOCLINIC TANGENCIES OF DISSIPATIVE DIFFEOMORPHISMS

A representative model of a return map near homoclinic bifurcation is studied. This model is the so-called fattened Arnold map, a diffeomorp hism of the annulus. The dynamics is extremely rich, involving periodi city, quasiperiodicity and chaos. The method of study is a mixture of analytic perturbation theory, numerical continuation, iteration to an attractor and experiments, in which the guesses are inspired by the th eory. In rum the results lead to fine-tuning of the theory. This appro ach is a natural paradigm for the study of complicated dynamical syste ms. By following generic bifurcations, both local and homoclinic, vari ous routes to chaos and strange attractors are detected. Here, particu larly, the 'large' strange attractors which wind around the annulus ar e of interest. Furthermore, a global phenomenon regarding Arnold tongu es is important. This concerns the accumulation of tongues on lines of homoclinic bifurcation. This phenomenon sheds some new light on the o ccurrence of infinitely many sinks in certain cases, as predicted by t he theory.