Bifurcation Theory by Melnikov method in fast slow system

Abstract EN

The Melmkov method for smooth dynamical systems is extended to be applicable to the non smooth one for nonlinear impact systems.

this paper deals with studying a new subject of a singularity perturbed ordinary differential equations system.

It is studied the ways to deal with the perturbation parameter > 0.

Then the bifurcation theory is applied on the last system according to singularity perturbed ODEs.

In addition, sufficient conditions for the occurrence of some types of bifurcation in the solution are given, such as (Fold, Pitchfork and Transcritical Bifurcation).

Depending on the proof of theories to reduce the singular perturbation ODEs.

for this purpose, proof of bifurcation that occurs in singular perturbation Teorem in this kind of situations is depended on the nature and behavior of the solution at the level of each state of bifurcation.