ABSTRACT
This chapter deals with the consequences of having a homoclinic orbit in different set-ups. For the theorist that is studying a natural system, the problems appear usually in a different way, namely that one has a model of a system that experimentally has shown (definite or strong) indications of being chaotic. An interesting phenomenon resembling a period-doubling bifurcation may occur for a suitable choice of parameter values, namely that a double-loop homoclinic orbit may branch from (or die at) a homoclinic curve. This mechanism allows for having more than one periodic orbit close (in parameter space) to the homoclinic orbit. The striking feature about the Šilnikov phenomenon is that it has more than just infinitely many periodic orbits. Homoclinic orbits in 2-D flows are quite featureless, unlike their counterparts in 2-D maps.
