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Periodic solutions of a planar delay equation

Published online by Cambridge University Press:  14 November 2011

Plácido Táboas
Plácido Táboas
Affiliation:
Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, RI 02912, U.S.A.; and (permanent address) ICMSC-USP Caixa Postal 668, 13560 Sã;o Carlos SP, Brasil
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Synopsis

We study the planar delay differential equation x′(t) = −x(t) + αF(x(t − 1)), for α > 0. An existence theorem for nonconstant periodic solutions is achieved for a certain class of maps F, for α > some α0. Besides a condition of nondegeneracy at x = 0, we assume F is bounded and satisfies a kind of planar negative feedback condition. The nonconstant periodic solutions are associated with nontrivial fixed points of a certain operator defined by the flow in the plase space C([−l, 0], R2). In our approach, the existence of such fixed points depends on the ejectivity of O ϵ C([−1, 0], R2) with respect to that operator. Relaxing the boundedness condition on F, we show the existence of a sequence of values of α, α0 < α1 <…, where a Hopf bifurcation occurs.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 116 , Issue 1-2 , 1990 , pp. 85 - 101
Copyright
Copyright © Royal Society of Edinburgh 1990

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