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Normal forms for periodic retarded functional differential equations

Published online by Cambridge University Press:  14 November 2011

Teresa Faria
Teresa Faria
Affiliation:
Departamento de Matemática, Faculdade de Ciências/CMAF, Univ. Lisboa, R. Ernesto Vasconcelos, 1700 Lisboa, Portugal
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Abstract

This paper addresses the computation of normal forms for periodic retarded functional differential equations (FDEs) with autonomous linear part. The analysis is based on the theory previously developed for autonomous retarded FDEs. Adequate nonresonance conditions are derived. As an illustration, the Bogdanov–Takens and the Hopf singularities are considered.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 1 , 1997 , pp. 21 - 46
Copyright
Copyright © Royal Society of Edinburgh 1997

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References

1Arnold, V. I.. Geometrical Methods in the Theory of Ordinary Differential Equations (New York: Springer, 1983).CrossRefGoogle Scholar
2Bogdanov, R. I.. Versal deformations of a singular point of a vector field on a plane in the case of zero eigenvalues. Functional Anal. Appl. 9 (1975), 144–5; 10 (1976), 61–2.CrossRefGoogle Scholar
3Bogdanov, R. I.. Orbital equivalence of singular points of vector fields on the plane. Functional Anal. Appl. 10(1976), 316–17.CrossRefGoogle Scholar
4Chow, S.-N. and Hale, J. K.. Methods of Bifurcation Theory (New York: Springer, 1982).CrossRefGoogle Scholar
5Chow, S.-N. and Mallet-Paret, J.. Integral averaging and bifurcation. J. Differential Equations 26 (1977), 112–59.CrossRefGoogle Scholar
6Faria, T. and Magalhaes, L. T.. Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity. J. Differential Equations 122 (1995), 201–224.CrossRefGoogle Scholar
7Faria, T. and Magalhaes, L. T.. Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation. J. Differential Equations 122 (1995), 181–200.CrossRefGoogle Scholar
8Gukenheimer, J. and Holmes, P.. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (New York: Springer, 1983).CrossRefGoogle Scholar
9Hale, J. K.. Theory of Functional Differential Equations (Berlin: Springer, 1977).CrossRefGoogle Scholar
10Hale, J. K. and Verduyn, S. M.Lunel. Averaging in infinite dimensions. J. Integral Equations Appl. 4 (1990), 463–94Google Scholar
11Henry, D.. The adjoint of a linear functional differential equation and boundary values problems. J. Differential Equations 9 (1971), 55–66.CrossRefGoogle Scholar
12Medved, M.. Two-sided solutions of linear integrodifferential equations of Voltera type with delay. Časopis Pěst. Mat. 155 (1990), 264–72.CrossRefGoogle Scholar
13Stech, A.. Hopf bifurcation calculations for functional differential equations. J. Math. Anal. Appl. 109 (1985), 472–91.CrossRefGoogle Scholar
14Takens, F.. Singularities of vector fields. Inst. Hautes Études Sci. Publ. Math. 43 (1974), 47–100.CrossRefGoogle Scholar