Existence of center for planar differential systems with impulsive perturbations (Q277831)

Consider the following nonlinear impulsive differential systems: NEWLINE\[NEWLINE\begin{aligned} \dot x= Ax+ f(x),\quad & x\not\in T_0,\\ \Delta x= Bx,\quad &x\in T_0,\end{aligned}\tag{1}NEWLINE\]NEWLINE where \(A=\left[\begin{matrix} \alpha &-\beta\\ \beta & \alpha\end{matrix}\right]\), \(\beta>0\); \(x=[x_1,x_2]^T\); and \(f(x)={f_1(x_1,x_2)\brack f_2(x_1,x_2)}\) is a real analytic vector function in the neighborhood of the origin such that \(f(x)= o(\| x\|)\) and \(f_i(0,0)= 0\), \(i= 1,2\); \(T_0=\bigcup^p_{i=1} s_i\), \(p\in\mathbb{N}^+\), where \(s_i\) are half-lines starting at the origin in the planar coordinate; \(B= \left[\begin{matrix} b_{11} & b_{12}\\ b_{21} & b_{22}\end{matrix}\right]\).NEWLINENEWLINE In the present paper, the method of solving of center-focus problem for (1) is specified.