On the simplest system with retarding switching and 2-point critical set.- Functional Differential Equations

TL;DR

This paper analyzes a two-equation system with retarding switching based on a critical set, characterizing solution behaviors for specific delay values, extending previous work to cover more complex dynamics.

Contribution

It provides a comprehensive characterization of solution behaviors for all delay values, especially in the previously unexamined interval (4/3, 3/2), revealing more complex dynamics.

Findings

Behavior of solutions characterized for τ in (4/3, 3/2)

Extended analysis to all τ > 0

Identified increased complexity in solution dynamics within certain delay ranges

Abstract

The system considered in this paper consists of two equations $(k=1,2)$ $\dot x(t)=(-1)^{k-1} (0\le t<\infty),\,k(0)=1,\,x(0)=0,\,x(t)\not\in\{0,1\}(-1\le t<0),$ that change mutually in every instant $t$ for which $x(t-\tau)\in\{0,1\}$, where $\tau={\rm const}>0$ is given. In this paper the behavior of the solutions is characterized for every $\tau\in(4/3, 3/2)$, i. e. in case not covered in \cite{ADM}; as it was noted there, this behavior turned out to be more complex then when $\tau\in(3/2,\infty)$. Thus the behavior of the solutions of this system with critical set $K=\{0,1\}$ is characterized for every $\tau>0$.