Morse Decomposition of Scalar Differential Equations with State-Dependent Delay

Bartha, Ferenc Ágoston and Garab, Ábel and Krisztin, Tibor (2025) Morse Decomposition of Scalar Differential Equations with State-Dependent Delay. JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS. ISSN 1040-7294

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Abstract

We consider state-dependent delay differential equations of the form \begin{aligned} \dot{x}(t) = f(x(t), x(t - r(x_t))), \end{aligned} x ˙ ( t ) = f ( x ( t ) , x ( t - r ( x t ) ) ) , where f is continuously differentiable and fulfills a negative feedback condition in the delayed term. Under suitable conditions on r and f , we construct a Morse decomposition of the global attractor, giving some insight into the global dynamics. The Morse sets in the decomposition are closely related to the level sets of an integer-valued Lyapunov function that counts the number of sign changes along solutions on intervals of length of the delay. This generalizes former results for constant delay. We also give two major types of state-dependent delays for which our results apply.

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