## Morse decompositions for delay-differential equations.(English)Zbl 0648.34082

The author deals with global dynamical properties of the scalar delay- differential equation (1) $$\dot x(t)=-f(x(t),x(t-1)),$$ where f satisfies appropriate conditions, in particular, a negative feedback condition in the delay. Main results of the paper are that the dynamical system for (1) possesses a global integer value Lypunov function which gives rise to a Morse decomposition of the attractor. These results are then used to obtain information about the asymptotic oscillatory properties of solutions to arbitrary initial value problems. The relation between periodic solutions of (1) and the Morse decomposition is also explored.
Reviewer: J.Ohriska

### MSC:

 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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### References:

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