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On the existence of periodic solutions to a system of two differential equations with pulse influence. (English. Russian original) Zbl 1015.34034
Ukr. Math. J. 54, No. 1, 160-165 (2002); translation from Ukr. Mat. Zh. 54, No. 1, 133-137 (2002).
The author studies the following 2-dimensional system: $\dot x=Jx \text{ while } \langle x,g\rangle +c>0;\quad x(t+0)-x(t-0)=h, \text{ if }\langle x(t-0),g\rangle +c=0.\tag{1}$ Here, $$\langle \cdot,\cdot \rangle$$ denotes the scalar product in the coordinate space $$\mathbb{R}^2$$, $$g,h\in \mathbb{R}^2$$ are given vectors satisfying the condition $$\langle g,h\rangle >0$$, $$c\in \mathbb{R}$$ is a given number, and $$J$$ is a $$2\times 2$$-dimensional Jordan matrix whose eigenvalues have negative real parts.
It is obvious that if, for a point $$x_0$$ belonging to the line $$L:=\{x\in \mathbb{R}^2:\langle x,g\rangle +c=0\}$$, there exists a $$T>0$$ such that $$e^{TJ}(x_0+h)=x_0$$ and $$\langle e^{TJ}(x_0+h),g\rangle +c>0$$, $$t\in (0,T)$$, then the point $$x_0$$ gives raise to a $$T$$-periodic solution to system (1). Basing on this fact, in order to find periodic solutions to (1), the author seeks sufficient conditions for the solvability of the system $$\{\langle x,g\rangle +c=0, x=e^{JT}(x+h)\}$$ with respect to the unknowns $$x,T$$. These conditions are expressed in terms of eigenvalues of $$J$$ and the angle between the vector $$h$$ and one of the basis vectors of $$\mathbb{R}^2$$.
##### MSC:
 34C25 Periodic solutions to ordinary differential equations 34A37 Ordinary differential equations with impulses
##### Keywords:
system with pulse action; periodic solution; fixed point
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