Spectral Theory for Perturbed Systems

Abstract

Key words perturbations, skew product flows, spectra MSC (2000) 37H10, 34D08, 93B05 This paper presents an overview of topological, smooth, and control techniques for dynamical systems and their interrelations for the study of perturbed systems. We concentrate on spectral analysis via linearization of systems. Emphasis is placed on parameter dependent perturbed systems and on a comparison of the Markovian and the dynamical structure of systems with Markov diffusion perturbation process. A number of applications is provided.

Stability reserve in stochastic linear systems with applications to power systems

“…This theorem was first proved in [3], the version presented here follows the set-up of [4]. The Lyapunov exponent from Theorem 2.1 determines the stability behavior of the solutions of (1) in the following way: Corollary 2.2: Under the conditions of Theorem 2.1, the zero solution ϕ(t, 0, ξ t ) ≡ 0 of the stochastic linear systeṁ x(t) = A(ξ t )x(t) is almost surely exponentially stable if and only if λ < 0.…”

“…Theorem 2.1: Consider the linear system (1) with stochastic perturbation (2,4) under the assumptions (3,7). Then the system has a unique Lyapunov exponent…”

“…For background material on Lyapunov exponents of stochastic systems we refer the reader to [1], [4] and [14].…”