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Flows on centre manifolds for scalar functional differential equations

Published online by Cambridge University Press:  14 November 2011

Jack K. Hale
Jack K. Hale
Affiliation:
Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, U.S.A
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Synopsis

By assuming that a linear scalar functional differential equation (FDE) has only the zero eigenvalue on the imaginary axis, it is shown that the flows on the centre manifolds of all Cr-perturbations of this equation coincide with the flows obtained from scalar ordinary differential equations (ODEs) of order m, where m is the multiplicity of the zero eigenvalue. Furthermore, it is shown that the above situation can be realized through differential difference equations with m – 1 fixed distinct delays.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 101 , Issue 3-4 , 1985 , pp. 193 - 201
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1Arnold, V. I.. Geometric Methods in the Theory of Ordinary differential equations (Grundlehren der Math. Wiss., Vol. 250) (Berlin: Springer, 1983).CrossRefGoogle Scholar
2Bogdanov, R. I.. Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues. Functional Anal. Appl. 9 (1975), 144–145; Ibid. 10 (1976), 61–62; Trudy Sem. Petrovsk. 2, 37–65.Google Scholar
3Hale, J. K.. Functional Differential Equations (Appl. Math. Sci., Vol. 3) (Berlin: Springer, 1977).CrossRefGoogle Scholar
4Lee, E. B. and Markus, L.. Foundations of Optimal Control Systems (New York: John Wiley, 1967).Google Scholar
5Lefschetz, S.. Stability of Nonlinear Control Systems (New York: Academic Press, 1965).Google Scholar