On the simplest system with retarding switching and 2-point critical set

Functional Differential Equations. 2004. Vol. 11. No. 3-4. P. 333–339.

Филимонов Д. А.

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 $\tau\in(\frac{4}{3},\frac{3}{2})$, 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 $\tau>0$.

Ключевые слова: functional differential equations, Periodic solutions, System with retarding switching