Kamal Hamid Yasir & Zahraa Ali Mutar

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

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

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 .

Slow-fast system

Normal hyperbolicity

Center Manifold

Fenichel's Theorem Fold bifurcation.

[1] J. Carr, "Applications of center manifold theory", Springer-Verlag, New York, (1981).

[2] P. Kokotovic, H. K. Khalil, and J. O'Reilly. "Singular Perturbation Methods in Control: Analysis and Design". London: Orlando Academic Press, (1986).

[3] R. L. Bishop, and R. J. Crittenden,"Geometry of Manifolds". New York, NY: Academic (1996).

[4] R. E. O'Malley, "Singular Perturbation Methods for Ordinary Differential Equations", Springer-Verlag, New York, (1991).

[5] Y. A. Kuznetsov,"Elements of Applied Bifurcation Theory", Second Edition, Springer-Verlag, New York, (1998).

[6] N. Fenichel,"Geometric singular perturbation theory for ordinary differential equations". J. Differential Equations, Vol. 31, No. 1, pp. 5398, 1979.

[7] C. Kuehn, "Multiple Time Scale Dynamics", Springer Cham Heidelberg New York Dordrecht London, (2015).

[8] A.Panfilov, S.Ma ee, "Non-linear dynamical systems", Utrecht University, Utrecht, (2005).

[9]A. Tikhonov. "On the dependence of the solutions of differential equations on a small

parameter", MatematicheskiiSbornik, pp.193-204. (1948).

[10] J. W. Riley, "Fenichel's theorems with applications in dynamical systems", University of Louisville, (2012).

[11] C. K. Jones,"Geometric singular perturbation theory in Dynamical Systems", Lecture Notes in Math, Springer-Verlag, Berlin, pp.44118. (1995).

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

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 systemsSlow-fast system

Normal hyperbolicity

Center Manifold

Fenichel's Theorem Fold bifurcation.

**References**

[1] J. Carr, "Applications of center manifold theory", Springer-Verlag, New York, (1981).

[2] P. Kokotovic, H. K. Khalil, and J. O'Reilly. "Singular Perturbation Methods in Control: Analysis and Design". London: Orlando Academic Press, (1986).

[3] R. L. Bishop, and R. J. Crittenden,"Geometry of Manifolds". New York, NY: Academic (1996).

[4] R. E. O'Malley, "Singular Perturbation Methods for Ordinary Differential Equations", Springer-Verlag, New York, (1991).

[5] Y. A. Kuznetsov,"Elements of Applied Bifurcation Theory", Second Edition, Springer-Verlag, New York, (1998).

[6] N. Fenichel,"Geometric singular perturbation theory for ordinary differential equations". J. Differential Equations, Vol. 31, No. 1, pp. 5398, 1979.

[7] C. Kuehn, "Multiple Time Scale Dynamics", Springer Cham Heidelberg New York Dordrecht London, (2015).

[8] A.Panfilov, S.Ma ee, "Non-linear dynamical systems", Utrecht University, Utrecht, (2005).

[9]A. Tikhonov. "On the dependence of the solutions of differential equations on a small

parameter", MatematicheskiiSbornik, pp.193-204. (1948).

[10] J. W. Riley, "Fenichel's theorems with applications in dynamical systems", University of Louisville, (2012).

[11] C. K. Jones,"Geometric singular perturbation theory in Dynamical Systems", Lecture Notes in Math, Springer-Verlag, Berlin, pp.44118. (1995).