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

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