Periodic solutions of a planar delay equation
- 1 January 1990
- journal article
- research article
- Published by Cambridge University Press (CUP) in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- Vol. 116 (1-2) , 85-101
- https://doi.org/10.1017/s0308210500031395
Abstract
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.Keywords
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