Homoclinic Bifurcation in the Parameter Space

In the end of the sixties, Shilnikov showed that the existence of a
homoclinic
orbit in a certain class of 3D flow implies the occurrence of
chaotic
behavior. This theorem was verified experimentally in glow-discharge
systems,
spiking neurons, rabbit arteries intermittency, electrochemical
oscillators,
among others. In the last two decades much work has been done to
understand
how the Shilnikov homoclinic orbits arise and are organized in the
parameter
space. In this talk, I will give an overview of this topic and present
new
results in the case of a 2-dimensional parameter space. I will also
present
numerical results of the Chua's circuit, where the prediction of the
theory
was verified.