The existence of periodic orbits for nonlinear impulsive differential systems
The authors consider a nonlinear autonomous impulsive differential system, \(x'= f(x)\), \(t\neq t_i\) \((t>0)\) under impulsive condition \(x(t_i+ 0)- x(t_i- 0)= I_i(x)\), where the impulses act in a set of fixed points \(\{t_i\}^{+\infty}_{i= 0}\), \(0= t_0< t_1<\cdots< t_n<\cdots\) (\(t_n\to+\infty\) as \(n\to+\infty\)). The functions \(f,I_i: \Omega\to \mathbb{R}\) \((\Omega\subset\mathbb{R})\) are continuous and \(x+ I_i(x)\in\Omega\) for \(x\in\Omega\) and every \(i\). The initial value is \(x(0)= x_0\). The necessary and sufficient condition for existence of periodic solutions is investigated. Stability properties are discussed as well. The main result is that the considered problem possesses a periodic orbit if \(G(x)= \tau+ \int^{h(x)}_x f^{-1}(s) ds\) has a zero point in \((0,+\infty)\) (\(h\in C(\mathbb{R}^+)\) is increasing in \(\mathbb{R}^+\) and \(h(0)= 0\)); \(\tau\) is the minimal positive period of any periodic orbit of the considered impulsive problem.
scientific article; zbMATH DE number 1359981