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

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.