Richness of dynamics and global bifurcations in systems with a
homoclinic figure-eight

V. Gonchenko 
Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod University,
Russia 

C. Sim\'o and A. Vieiro 
Dpt. Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona, Spain

Abstract
We consider 2D flows having a homoclinic figure-eight to a dissipative saddle.
We study the rich dynamics that such a system exhibits under a periodic forcing.
First, we derive the bifurcation diagram using topological techniques. In
particular, there is a homoclinic zone in the parameter space which has a
non-smooth boundary. We provide a complete explanation of this phenomenon
relating it to primary quadratic homoclinic tangency curves which end up at some
cubic tangency (cusp) points. We also describe the possible attractors that
exist (and may coexist) in the system. A main goal of this work is to show how
the previous qualitative description can be complemented with quantitative
global information. To this end, we introduce a return map model which can be
seen as the simplest one which is ``universal'' in some sense. We carry out
several numerical experiments on the model, to check that all the objects
predicted to exist by the theory are found in the model, and also to investigate
new properties of the system.