Morse decomposition of scalar differential equations with state-dependent delay

TL;DR

This work analyzes scalar delay differential equations with state-dependent delays under negative feedback and dissipativity, establishing a Morse decomposition of the global attractor. It develops a nonautonomous linearization and an integer-valued, discrete Lyapunov function that counts sign changes on delay intervals, enabling a level-set-based organization of global dynamics analogous to constant-delay results. A key contribution is the construction of Morse sets from linearized spectral data and sign-change levels, including a special+$\mathcal{S}_{N_0}^+$ class to accommodate high-frequency oscillations when $V$ is unbounded. Under an additional iterated-zeros condition, the Lyapunov function is bounded, yielding a finite Morse decomposition and sharper results for two representative state-dependent delay classes (threshold-type and two-value delays). Overall, the paper extends Morse-decomposition techniques to state-dependent delays and provides concrete delay classes where the theory yields tangible, bounded stratifications of global dynamics.

Abstract

We consider state-dependent delay differential equations of the form $$\dot{x}(t) = f(x(t), x(t - r(x_t))),$$ 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.

Figures (2)

• Figure 3.1: Illustration of the proof of \ref{['korlatozo-v']}. On the right-hand side, we see how we can find two values ($s_{1,0}$ and $s_{1,1}$), using part (i) and the mean value theorem, where the derivatives have opposite signs. This results in a sign change of $y$ between $t_{1,0}=\eta(s_{1,0})$ and $t_{1,1}=\eta(s_{1,1})$. The latter is illustrated on the left-hand side of the figure. Then the procedure is continued to obtain 3 values ($s_{2,0}$, $s_{2,1}$ and $s_{2,2}$) with interchanging signs of derivatives of $y$.
• Figure 5.1: Example of an entire solution not obeying condition \ref{['iterated_zeros_imply_fullzero']}.

Theorems & Definitions (48)

• Proposition 2.1
• proof
• Remark 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Lemma 2.5
• proof
• Lemma 3.1: krisztin-arino
• Lemma 3.2: krisztin-arino