Abstract
Transition layers arising from square-wave-like periodic solutions of a singularly perturbed delay differential equation are studied. Such transition layers correspond to heteroclinic orbits connecting a pair of equilibria of an associated system of transition layer equations. Assuming a monotonicity condition in the nonlinearity, we prove these transition layer equations possess a unique heteroclinic orbit, and that this orbit is monotone. The proof involves a global continuation for heteroclinic orbits.
Similar content being viewed by others
References
Chow, S.-N., Diekmann, O., and Mallet-Paret, J. (1985). Stability, multiplicity, and global continuation of symmetric periodic solutions of a nonlinear Volterra integral equation.Jpn. J. Appl. Math. 2, 433–469.
Chow, S.-N., and Mallet-Paret, J. (1983). Singularly perturbed delay-differential equations.North Holland Math. Stud. 80, 7–12.
Hale, J. (1977).Theory of Functional Differential Equations, Springer-Verlag, New York.
Hale, J., and Lin, X.-B. (1986). Heteroclinic orbits for retarded functional differential equations.J. Differential Equations 65, 175–202.
Hartman, P. (1964).Ordinary Differential Equations, John Wiley and Sons, New York.
Lin, X.-B. (1986). Exponential dichotomies and homoclinic orbits in functional differential equations.J. Differential Equations 63, 227–254.
Mallet-Paret, J. (1988). Morse decompositions for delay-differential equations.J. Differential Equations 72, 270–315.
Mallet-Paret, J., and Nussbaum, R. D. (1986a). Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation.Ann. Mat. Pura Appl. 145, 33–128.
Mallet-Paret, J., and Nussbaum, R. D. (1986b). Global continuation and complicated trajectories for periodic solutions for a differential-delay equation.Proc. Symp. Pure Math. Am. Math. Soc. [Part 2],45, 155–167.
Palmer, K. J. (1984). Exponential dichotomies and transversal homoclinic points.J. Differential Equations 55, 225–256.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chow, SN., Lin, XB. & Mallet-Paret, J. Transition layers for singularly perturbed delay differential equations with monotone nonlinearities. J Dyn Diff Equat 1, 3–43 (1989). https://doi.org/10.1007/BF01048789
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01048789