A Lyapunov and Sacker–Sell spectral stability theory for one-step methods

Theory and Methods for One -step Odes

where f is continuous and outputs an Rd vector. approximation on grid functions Let y(t) be the true solution. LetH be the grid dened by step-sizes: h1 , . . . , hN such that 0 < t1 < . . . < tN satises t i = ∑ l=1 h i .1 Let ∣H∣ = maxi h i . 1 is is just the obvious grid where h i = t i − t i−1 . We are parameterizing it by the step-sizes h i instead of times because h is what shows up in th...

Lyapunov Spectral Intervals: Theory and Computation

Different definitions of spectra have been proposed during the years to characterize the asymptotic behavior of nonautonomous linear systems. Here, we consider the spectrum based on exponential dichotomy of Sacker and Sell and the spectrum defined in terms of upper and lower Lyapunov exponents. A main goal of ours is to understand to what extent these spectra are computable. By using an orthogo...

Duality in Stability Theory: Lyapunov Function and Lyapunov Measure

In this paper, we introduce Lyapunov measure to verify weak (a.e.) notions of stability for an invariant set of a nonlinear dynamical system. Using certain linear operators from Ergodic theory, we derive Lyapunov results to deduce a.e. stability. In order to highlight the linearity of our approach, we also provide explicit formulas for obtaining Lyapunov function and Lyapunov measure in terms o...

Inverse Spectral Theory for One

Fuzzy Lyapunov stability and exponential stability in control ‎systems‎

Fuzzy control systems have had various applications in a wide range of science and engineering in recent years. Since an unstable control system is typically useless and potentially dangerous, stability is the most important requirement for any control system (including fuzzy control system). Conceptually, there are two types of stability for control systems: Lyapunov stability (a special case ...