15:00 - 16:30, November 27, 2024 (Wednesday) Cisco Webex, Online seminar
(線上演講 Cisco Webex)
Cyclic System of Differential Equations with Constant, Time-variable or State-dependent Delay Ábel Garab (University of Szeged)
Abstract
The talk is based on joint work with Ferenc A. Bartha, István Balázs and Tibor Krisztin (University of Szeged) [1-3].
Consider nonautonomous cyclic systems of delay differential equations of the form
with time variable delays and where the coordinates are meant modulo . Such equations arise in many applications such as position control, engineering problems or various biological models.
Under suitable assumptions -- including a feedback condition in the third variable -- we define an integer valued Lyapunov functional, related to the number of sign-changes on certain intervals of the coordinate functions of solutions, and review some of its properties that are known from the constant delay case [4] and from the scalar case (i.e. ) with negative feedback [5].
In the autonomous case we use these properties to prove that if the delays are constant, the global attractor has a gradient-like structure. We derive analogous results in the scalar case (i.e. ) with rather generic state-dependent delay.
Reference
[1] I. Balázs, Á. Garab, Discrete Lyapunov functional for a cyclic system of differential equations with state-dependent or variable delay, preprint.
[2] F. A. Bartha, Á. Garab, T. Krisztin, Morse decomposition for state-dependent delay differential equations, submitted. https://arxiv.org/abs/2410.23491
[3] Á. Garab Absence of small solutions and existence of Morse decomposition for a cyclic system of delay differential equations, J. Differential Equations 269( 2020), No. 6, 5463--5490.
[4] J. Mallet-Paret, G. R Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, (1996), No. 2, 385–440.
[5] T. Krisztin, O. Arino, The two-dimensional attractor of a differential equation with state-dependent delay, (2001), No. 3, 453-522.
