Transversal heteroclinic and homoclinic orbits in singular perturbation problems

Abstract In this paper we give a geometric construction of heteroclinic and homoclinic orbits for singularly perturbed differential equations. By using methods from invariant manifold theory we show that transversal intersection of stable and unstable manifolds of the reduced problem implies the existence of transversal heteroclinic or homoclinic orbits of the singularly perturbed problem. We derive analytical conditions for transversality. We show how these results can be used to prove the existence of heteroclinic and homoclinic orbits in singularly perturbed problems which depend on additional parameters. We describe a configuration which implies transversal intersection of the stable and unstable manifolds of periodic orbits and the associated chaotic dynamics.

[1]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[2]  F. Hoppensteadt.  Properties of solutions of ordinary differential equations with small parameters , 1971 .

[3]  C. Schmeiser,et al.  Asymptotic analysis of singular singularly perturbed boundary value problems , 1986 .

[4]  Kenneth J. Palmer,et al.  Exponential dichotomies and transversal homoclinic points , 1984 .

[5]  Xiaobiao Lin.  Shadowing lemma and singularly perturbed boundary value problems , 1989 .

[6]  Neil Fenichel.  Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[7]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[8]  Jack K. Hale,et al.  Existence and stability of transition layers , 1988 .

[9]  G. Carpenter.  A geometric approach to singular perturbation problems with applications to nerve impulse equations , 1977 .

[10]  Shui-Nee Chow,et al.  An example of bifurcation to homoclinic orbits , 1980 .

[11]  H. McKean.  Nagumo's equation , 1970 .

[12]  The bifurcations of countable connections from a twisted heteroclinic loop , 1991 .

[13]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[14]  David Terman,et al.  Propagation Phenomena in a Bistable Reaction-Diffusion System , 1982 .

[15]  S. Hastings.  Single and Multiple Pulse Waves for the FitzHugh–Nagumo , 1982 .

[16]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[17]  Xiao-Biao Lin,et al.  Heteroclinic bifurcation and singularly perturbed boundary value problems , 1990 .

[18]  P. Fife.  Transition layers in singular perturbation problems , 1974 .