Bifurcation by Application Fenichel's Theorem to Singularity Perturbed ODEs

Kamal Hamid Yasir1 , Zahraa Ali Mutar1

1 Department of Mathematics, College of Education for Pure Science, Thi-Qar University,  Iraq

Original: 20 May 2016, Revised: 29 August 2016, Accepted: 16 October 2016, Published online: 20 March 2017

DOI: https://doi.org/10.17656/jzs.10595

Abstract

In this Paper, we will state a special structure of the singularly perturbed system with changes in time scales, when the perturbation parameter . That led to a new structure of fast system. In addition, we will study and studying the invariant manifold of flow of the vector field, and normal hyperpolicity of the fast-slow system in order to provide a proof of the perturbation of normally hyperbolic invariant manifolds due to Fenichel's Theorem.

So, we will introduce some basic ideas of the general case of fast-slow system. Also, we introduce an invariant manifold theory that explain the concept of the normal hyperbolicity invariant manifold of fast-slow system when . Center Manifold Theorem on fast-slow system will have been stated in order to get an invariant manifold of singularity perturbed ODEs system, also, we will mention the connection between the Fenichel's Theorem and the application of the Center Manifold Theorem on fast-slow system, and then we will study bifurcation theory on singularity perturbed ODE when perturbed parameter .

Key Words:

Singularly perturbed systems

Slow-fast system

Normal hyperbolicity

Center Manifold

Fenichel's Theorem Fold bifurcation.

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