Linearization of impulsive differential equations with ordinary dichotomy
Summary: This paper presents a linearization theorem for the impulsive differential equations when the linear system has ordinary dichotomy. We prove that when the linear impulsive system has ordinary dichotomy, the nonlinear system \(\dot{x}(t) = A(t) x(t) + f(t, x)\), \(t \neq t_k\), \(\Delta x(t_k) = \widetilde{A}(t_k) x(t_k) + \widetilde{f}(t_k, x)\), \(k \in \mathbb Z\), is topologically conjugated to \(\dot{x}(t) = A(t) x(t)\), \(t \neq t_k\), \(\Delta x(t_k) = \widetilde{A}(t_k) x(t_k)\), \(k \in \mathbb Z\), where \(\Delta x(t_k) = x(t_k^+) - x(t_k^-)\), \(x(t_k^-) = x(t_k)\), represents the jump of the solution \(x(t)\) at \(t = t_k\). Finally, two examples are given to show the feasibility of our results.
scientific article; zbMATH DE number 7022783