This work is devoted to the stability of random-switching systems of differential equations. After presenting the formulation of random-switching systems, the notion of stability is recalled, and sufficient conditions in terms of the Liapunov function are presented. Then easily verifiable conditions for stability and instability of systems arising in approximation are established. Using a logar...

Given a smooth closed manifold M , the Morse-Witten complex associated to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology ofM . The geometric approach presented here was developed in [We-93] and is based on...

We present a new version of the Grobman-Hartman's linearization theorem for random dynamics. Our result holds infinite dimensional systems whose linear part is not necessarily invertible. In addition, by adding some restrictions on non-linear perturbations, we don't require to be nonuniformly hyperbolic in sense Pesin but rather (besides requiring existence stable and unstable directions) allow...

Discretizations and Grobman-Hartman Lemma, discretizations and the hierarchy of invariant manifolds about equilibria are considered. For one-step methods, it is proved that the linearizing conjugacy for ordinary differential equations in Grobman-Hartman Lemma is, with decreasing stepsize, the limit of the linearizing conjugacies of the discrete systems obtained via time-discretizations. Similar...

Due to the nonlinear and nonminimum phase characteristics of buck-boost converter, design its controller has always been a challenging problem. In this paper, multi-index feedback linearization control strategy is proposed converter. Firstly, by constructing an appropriate output function, original system mapped into combination linear subsystem subsystem. Then, according structural these two s...

This manuscript provides an introduction to ordinary differential equations and dynamical systems. We start with some simple examples of explicitly solvable equations. Then we prove the fundamental results concerning the initial value problem: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore we consider linear equations, the Floquet theorem, and the autonomous...