# Dichotomies and Stability in Nonautonomous Linear Systems (Stability and Control: Theory, Methods and Applications, 14)

by Yu. A. Mitropolsky

Publisher: CRC

Written in English

## Subjects:

• Applied mathematics,
• Differential Equations,
• Control Theory,
• Mathematics,
• Science/Mathematics,
• Applied,
• Number Systems,
• Mathematics / General,
• Differentiable Dynamical Systems,
• Differentiable dynamical syste,
• Differential equations, Linear,
• Stability
The Physical Object
FormatHardcover
Number of Pages368
ID Numbers
Open LibraryOL9869759M
ISBN 100415272211
ISBN 109780415272216

Nonuniform Exponential Dichotomies and Lyapunov Regularity The more general notion of nonuniform exponential dichotomy plays a similar role under much weaker assumptions. On the other hand, for nonautonomous linear equations v′ = A(t)v with global solutions, we show here that this more general notion is in fact as weak as possible: namely. stability for the linear equation, although this requirement is dramatically restrictive for a nonautonomous system. Incidentally, this assumption is analogous to the restrictive requirement of existence of an exponential dichotomy for the evolution operator of a nonautonomous equation in the case when there exist simultaneously positive and. This book contains a systematic exposition of the elements of the asymptotic stability theory of general non-autonomous dynamical systems in metric spaces with an emphasis on the application for different classes of non-autonomous evolution equations (Ordinary Differential Equations (ODEs), Difference Equations (DEs), Functional-Differential. L. Barreira and C. Valls, Robustness of nonuniform exponential dichotomies in Banach spaces, J. Differential Equations, (), – \ref\key 3 ––––, Stability of Nonautonomous Differential Equations, Lecture Notes in Mathematics, , Springer () \ref\key 4 ––––, Polynomial contractions and Lyapunov regularity Cited by:

On (,)-Dichotomies for Nonautonomous Linear Difference Equations in Banach Spaces MihaiGabrielBabu uia,1 MihailMegan, 1,2 andIoan-LucianPopa 1 Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timis ¸oara, V. P .   This invaluable book studies synchronization of coupled chaotic circuits and systems, as well as its applications. It shows how one can use stability results in nonlinear control to derive synchronization criteria for coupled chaotic circuits and systems. It also discusses the use of Lyapunov exponents in deriving synchronization criteria. j=1 be solutions to linear system (). If their Wronskian is equal to zero at least at one point t0 ∈ Ithen these vector functions are linearly dependent. Proof. (a) and (b) are the consequences of the standard facts from linear algebra and left as exercises. To show (c), assume that t0 is such that W(t0) = 0. It means that the linear File Size: KB. Discrete & Continuous Dynamical Systems - A, , 38 (9): doi: /dcds [19] A. Rodríguez-Bernal. Perturbation of the exponential type of linear nonautonomous parabolic equations and applications to nonlinear : Yongxin Jiang, Can Zhang, Zhaosheng Feng.

differentiable” N ×N autonomous system of differential equations. However, since we are beginners, we will mainly limit ourselves to 2×2 systems. The Systems of Interest and a Little Review Our interest in this chapter concerns fairly arbitrary 2×2 autonomous systems of differential equations; that is, systems of the form x′ = f (x, y)File Size: KB. We generalize the theory of underlying one-step methods to strictly stable general linear methods (GLMs) solving nonautonomous ordinary differential equations (ODEs) that satisfy a global Lipschitz condition. We combine this theory with the Lyapunov and Sacker-Sell spectral stability theory for one-step methods developed in [ Find many great new & used options and get the best deals for Lecture Notes in Mathematics: Geometric Theory of Discrete Nonautonomous Dynamical Systems by Christian Pötzsche (, Paperback) at the best online prices at eBay! Free shipping for many products! The book treats the theory of attractors for non-autonomous dynamical systems. The aim of the book is to give a coherent account of the current state of the theory, using the framework of processes to impose the minimum of restrictions on the nature of the non-autonomous dependence. The book is intended as an up-to-date summary of the field, but much of it will be accessible to beginning.

## Dichotomies and Stability in Nonautonomous Linear Systems (Stability and Control: Theory, Methods and Applications, 14) by Yu. A. Mitropolsky Download PDF EPUB FB2

Dichotomies and Stability in Nonautonomous Linear Systems (Stability and Control: Theory, Methods and Applications) 1st Edition by Yu. Mitropolsky (Author), A.M. Samoilenko (Author), V.L. Kulik (Author) & 0 moreCited by: 1st Edition Published on Octo by CRC Press Linear nonautonomous equations arise as mathematical models in mechanics, chemistry, and biology.

The inv Dichotomies and Stability in Nonautonomous Linear Systems - 1st Editio. Dichotomies and Stability in Nonautonomous Linear Systems - CRC Press Book Linear nonautonomous equations arise as mathematical models in mechanics, chemistry, and biology.

The investigation of bounded solutions to systems of differential equations involves some important and challenging problems of perturbation theory for invariant toroidal manifolds. Linear nonautonomous equations arise as mathematical models in mechanics, chemistry, and biology.

The investigation of bounded solutions to systems of differential equations involves some important and challenging problems of perturbation theory for invariant toroidal manifolds. This monograph Price: $Dichotomies and Stability in Nonautonomous Linear Systems by Y. Mitropolskii,available at Book Depository with free delivery worldwide.3/5(1). Linear non-autonomous equations arise as mathematical models in mechanics, chemistry, and biology. This book explores the preservation of invariant tori of dynamic systems under perturbation. It is a useful contribution to the literature on stability theory and provides a source of reference for postgraduates and researchers. Linear nonautonomous equations arise as mathematical models in mechanics, chemistry, and biology. The investigation of bounded solutions to systems of differential equations involves some important and challenging problems of perturbation theory for invariant toroidal manifolds. This monograph is a detailed study of the application of Lyapunov functions with variable sign, expressed in. Linear non-autonomous equations arise as mathematical models in mechanics, chemistry, and biology. This book explores the preservation of invariant tori of dynamic systems under perturbation. for a class of linear nonautonomous systems. uction. The classical Liapunov approach to the study of asymptotic stability of an equilibrium of autonomous di erential equations relies on the exis-tence of a positive de nite Liapunov function with negative de nite time deriva-tive. been investigations of nonautonomous diﬀerential equations, that is with time-dependent vectorﬁelds, during this time, but it is only in Dichotomies and Stability in Nonautonomous Linear Systems book recent decade that a theory of nonautonomous dynamical systems has emerged synergizing parallel developments on time-dependent diﬀerential equations, control systems and ran-dom dynamical systems. Dichotomies and Stability in Nonautonomous Linear Systems, Stability and Control: Theory, Methods and Applications, vol. 14, Taylor & Francis () Google Scholar [19]Cited by: 2. Main theme of this volume is the stability of nonautonomous differential equations, with emphasis on the Lyapunov stability of solutions, the existence and smoothness of invariant manifolds, the. Concerning a necessary linear theory, our hyperbolicity concept is based on exponential dichotomies and splittings. This concept is in turn used to construct nonautonomous invariant manifolds, so-called fiber bundles, and deduce linearization theorems. Main theme of this volume is the stability of nonautonomous differential equations, with emphasis on the Lyapunov stability of solutions, the existence and smoothness of invariant manifolds, the construction and regularity of topological conjugacies, the study of center manifolds, as well as their reversibility and equivariance by: Abstract This paper considers two general concepts of dichotomy for noninvertible and nonautonomous linear discrete-time systems in Banach concepts use two types of dichotomy. The main theme of this book is the stability of nonautonomous di?erential equations, with emphasis on the study of the existence and smoothness of invariant manifolds, and the Lyapunov stability of solutions. We always c- sider a nonuniform exponential behavior of the linear variational equations, given by the existence of a nonuniform exponential contraction or a nonu- form exponential dichotomy. Main theme of this volume is the stability of nonautonomous differential equations, with emphasis on the Lyapunov stability of solutions, the existence and smoothness of invariant manifolds, the construction and regularity of topological conjugacies, the study of center manifolds, as well as their reversibility and equivariance properties. THE BOHL SPECTRUM FOR LINEAR NONAUTONOMOUS DIFFERENTIAL EQUATIONS THAI SON DOAN, KENNETH J. PALMER, AND MARTIN RASMUSSEN Dedicated to the memory of George R. Sell Abstract. We develop the Bohl spectrum for nonautonomous linear diﬀerential equation on a half line, which is a spectral concept that lies between the Lyapunov and the Sacker–Sell. In this paper we study the robustness of the stability in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. Conditions to establish Mittag-Leffler stability of solutions for nonlinear nonautonomous discrete Caputo-like fractional systems just from the linear associated system is shown. Mittag-Leffler stability for linear systems is tackled pointing out properties the matrix must : Luis Franco-Pérez, Guillermo Fernández-Anaya, Luis Alberto Quezada-Téllez. In the third section, the different flows induced by a family of linear Hamiltonian systems varying over a compact metric space (which usually arises in a natural way from a nonautonomous system Cited by: 3. Nonautonomous dynamical systems provide a mathematical framework for temporally changing phenomena, where the law of evolution varies in time due to seasonal, modulation, controlling or even random effects. Our goal is to provide an approach to the corresponding geometric theory of nonautonomous. Stability in Nonautonomous Dynamics this simpliﬁcation allows us to describe the main ideas without accessory technicalities. Consider the linear equation (1) in a Banach space X, for a family of bounded linear operators A(t) varying continuously with t ≥ 0. We assume that (1) has unique solutions that are deﬁned for all time t ≥ 0 Cited by: 1. This paper considers two general concepts of dichotomy for noninvertible and nonautonomous linear discrete-time systems in Banach spaces. These concepts use two types of dichotomy projections sequences (invariant and strongly invariant) and generalize some well-known dichotomy concepts (uniform, nonuniform, exponential, and polynomial). In the particular case of strongly invariant Cited by: For nonautonomous linear equations x′ = A(t)x, we give a complete characterization of the existence of exponential behavior in terms of Lyapunov functions. In particular, we obtain an inverse theorem giving explicitly Lyapunov functions for each exponential dichotomy. The main novelty of our work is that we consider a very general type of nonuniform exponential by: Summary Nonlinear systems with random structures arise quite frequently as mathematical models in diverse disciplines. This monograph presents a systematic treatment of stability theory and the theory of stabilization of nonlinear systems with random structure in terms of new developments in the direct Lyapunov's method. This paper considers two general concepts of dichotomy for noninvertible and nonautonomous linear discrete-time systems in Banach spaces. These concepts use two types of dichotomy projections sequences (invariant and strongly invariant) and generalize some well-known dichotomy concepts (uniform, nonuniform, exponential, and polynomial).Cited by: Dynamical Systems and Control 1st Edition. Edited by Firdaus E. Udwadia, H.I. Weber, George Leitmann. The 11th International Workshop on Dynamics and Control brought together scientists and engineers from diverse fields and gave them a venue to develop a greater understanding of this discipline and how it relates to many areas in science, engineering, economics, and biology. Stability and asymptotic estimates in nonautonomous linear differential systems Citation for published version (APA): Söderlind, G., & Mattheij, R. In this paper we consider some concepts of exponential splitting for nonautonomous linear discrete-time systems. These concepts are generalizations of some well-known concepts of (uniform and nonuniform) exponential dichotomies. Connections between these concepts are presented and some illustrating examples prove that these are distinct. We study the differentiability properties of the topological equivalence between a uniformly asymptotically stable linear nonautonomous system and a perturbed system with suitable nonlinearities. Dichotomies in stability theory. Lecture Notes in Mathematics (Berlin: Springer, ).Since then, many researchers have done further studies on the stability of linear fractional differential systems (see [7, 8]). In nonlinear systems, Lyapunov’s direct method provides a way to analyze the stability as Mittag-Leffler stability of a system without explicitly solving the differential equations (see [9–11]). The method Cited by: 3.In this paper, a new notion called the general nonuniform$(h,k,\mu,\nu)\$-dichotomy for a sequence of linear operators is proposed, which occurs in a more.