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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
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