Abstract
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.
Similar content being viewed by others
REFERENCES
J. K. Hale, Theory of Functional Differential Equations, Springer, New York (1977).
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).
I. I. Klevchuk, “Bifurcation of an equilibrium of a singularly perturbed system with delay,” Ukr. Mat. Zh., 47, No. 8, 1022–1028 (1995).
V. K. Mel'nikov, “On the stability of a center under perturbations periodic in time,” Tr. Mosk. Mat. Obshch., 12, 3–52 (1963).
K. J. Palmer, “Exponential dichotomies and transversal homoclinic points,” J. Different. Equat., 55, No. 2, 225–256 (1984).
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).
Z. Nitecki, Differential Dynamics [Russian translation], Mir, Moscow (1975).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Klevchuk, I.I. Homoclinic Points for a Singularly Perturbed System of Differential Equations with Delay. Ukrainian Mathematical Journal 54, 693–699 (2002). https://doi.org/10.1023/A:1021047730635
Issue Date:
DOI: https://doi.org/10.1023/A:1021047730635