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.