Homoclinic Points for a Singularly Perturbed System of Differential Equations with Delay
We obtain a representation of the integral manifold of a system of singularly perturbed differential-difference equations with periodic right-hand side. We show that, under certain conditions imposed on the right-hand side, the Poincaré map for the perturbed system has a transversal homoclinic point.
KeywordsDifferential Equation Integral Manifold Singularly Perturb Homoclinic Point Perturb System
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- 1.J. K. Hale, Theory of Functional Differential Equations, Springer, New York (1977).Google Scholar
- 2.Yu. A. Mitropol'skii, V. I. Fedchuk, and I. I. Klevchuk, “Integral manifolds, stability, and bifurcation of solutions of singularly perturbed functional differential equations,” Ukr. Mat. Zh., 38, No. 3, 335–340 (1986).Google Scholar
- 3.I. I. Klevchuk, “Bifurcation of an equilibrium of a singularly perturbed system with delay,” Ukr. Mat. Zh., 47, No. 8, 1022–1028 (1995).Google Scholar
- 4.V. K. Mel'nikov, “On the stability of a center under perturbations periodic in time,” Tr. Mosk. Mat. Obshch., 12, 3–52 (1963).Google Scholar
- 5.K. J. Palmer, “Exponential dichotomies and transversal homoclinic points,” J. Different. Equat., 55, No. 2, 225–256 (1984).Google Scholar
- 6.A. M. Samoilenko, O. Ya. Timchishin, and A. K. Prikarpatskii, “ The Poincaré-Mel'nikov geometric analysis of the transversal splitting of manifolds for slowly perturbed nonlinear dynamical systems. I,” Ukr. Mat. Zh., 45, No. 12, 1668–1681 (1993).Google Scholar
- 7.Z. Nitecki, Differential Dynamics [Russian translation], Mir, Moscow (1975).Google Scholar