Connections between Morse sets for delay-differential equations. (English) Zbl 0659.34077

Fiedler, Bernold; Maller-Paret, John

J. Reine Angew. Math. 397, 23-41 (1989).

The paper investigates the global dynamics of the scalar delay- differential equation \(x'(t)=-f(x(t),x(t-1))\) with appropriate conditions on f. In particular, a negative feedback condition in the delay term is assumed, which makes solutions tend to oscillate about the origin 0. The rate of oscillation is measured by an integer valued Lyapunov functional V, inducing a Morse decomposition. Results on the large time behavior of V on the global unstable manifold \(W^ u\) of 0 are obtained by topological arguments. Specifically, let \(N<\dim W^ u\) be an odd integer. Then \(W^ u\) contains a solution x(t) which has precisely N zeros, all simple, in any interval \(t\in [\tau -1,\tau]\) provided \(x(\tau)=0\) and \(\tau\) is chosen large enough. The solution x(t) is a heteroclinic connection between Morse sets. Further results investigate the dimensionality of sets of connecting orbits.

MSC:

34K99	Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
37D15	Morse-Smale systems

Keywords:

global dynamics; scalar delay-differential equation; Lyapunov functional; Morse decomposition; Morse sets