Morse decompositions and global continuation of periodic solutions for singularly perturbed delay equations. (English) Zbl 0562.34060

Mallet-Paret, John

Systems of nonlinear partial differential equations, Proc. NATO Adv. Study Inst., Oxford/U.K. 1982, NATO ASI Ser., Ser. C 111, 351-365 (1983).

[For the entire collection see Zbl 0514.00014.]
The class of differential delay equations (1) \(\sigma \dot x(t)=- x(t)+f(x(t-1))\) is considered where \(x\in R\), \(\sigma >0\), f is \(C^{\infty}\); \(f(0)=0\), \(f'(0)<-1\), \(xf(x)<0\) for all \(x\neq 0\) and \(| f(x)| <| x|\) for large \(| x|\). The author shows that there exists a global continuation of the Hopf bifurcation orbits of (1), maximal in a certain sense, and that for all \(\sigma\) (1) possesses Morse decomposition.

Reviewer: R.G.Koplatadze

MSC:

34K99	Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C25	Periodic solutions to ordinary differential equations

Keywords:

differential delay equations; Hopf bifurcation; Morse decomposition

Citations:

Zbl 0514.00014