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

Abstract

The system considered in this paper consists of two equations $(k=1,2)$ $\dot{x}(t)=(-1{)}^{k-1}(0\le t<\infty ),k(0)=1,x(0)=0,x(t)\notin \{0,1\}(-1\le t<0),$ that change mutually in every instant $t$ for which $x(t-\tau )\in \{0,1\}$, where $\tau =\mathrm{const}>0$ 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 $\tau >0$.

7 pages