Existence and robustness of nonuniform \((h,k,\mu ,\nu )\)-dichotomies for nonautonomous impulsive differential equations
The authors consider the linear impulsive differential equation in \(\mathbb{R}^n\) \[ x'=A(t) x, \quad t\geq 0, \;t\neq \tau_i, \qquad \triangle x|_{t=\tau_i}=B_i x, \;i\in \mathbb{N}, \] where \(A(t)\) is a continuous \(n\times n\) matrix function for each \(t\geq 0\), \(B_i\) is an \(n\times n\) matrix for each \(i\in \mathbb{N}\) and \(\{\tau_i \}_{i=1}^\infty\) is a sequence of numbers \[ 0< \tau_1< \tau_2<\dotsb <\tau_i< \dotsb, \quad \lim_{i\to \infty} \tau_i=\infty. \] They propose a notion of the nonuniform \((h,k,\mu,\nu)\)-dichotomy, which extends the notion of dichotomy of uniform and nonuniform types and is related to the theory of nonuniform hyperbolicity. They establish a sufficient criterion for the existence of the nonuniform \((h,k,\mu,\nu)\)-dichotomy in terms of appropriate Lyapunov exponents. For the linear perturbation they develop the robustness or roughness of nonuniform \((h,k,\mu,\nu)\)-dichotomies in the sense that a nonuniform \((h,k,\mu,\nu)\)-dichotomy persists under a sufficiently small linear perturbation.
nonuniform \((h,k,\mu ,\nu )\)-dichotomy
scientific article; zbMATH DE number 6143246