This volume covers the stability of nonautonomous differential equations in Banach spaces in the presence of nonuniform hyperbolicity. Topics under discussion include the Lyapunov stability of solutions, the existence and smoothness of invariant manifolds, and the construction and regularity of topological conjugacies. The exposition is directed to researchers as well as graduate students interested in differential equations and dynamical systems, particularly in stability theory.
The foundation of the modern theory of stability was created in the works of A Poincare and A M Lyapunov. The theory of the stability of motion has gained increasing significance in the last decade as is apparent from the large number of publications on the subject. A considerable part of these works are concerned with practical problems, especially problems from the area of controls and servo-mechanisms, and concrete problems from engineering, which first gave the decisive impetus for the expansion and modern development of stability theory. 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 Equations (FDEs), Semi-Linear Parabolic Equations etc). The basic results of this book are contained in the courses of lectures which the author has given during many years for the students of the State University of Moldova.This book is intended for mathematicians (scientists and university professors) who are working in the field of stability theory of differential/difference equations, dynamical systems and control theory. It would also be of use for the graduate and post graduate student who is interested in the theory of dynamical systems and its applications. The reader needs no deep knowledge of special branches of mathematics, although it should be easier for readers who know the fundamentals concepts of the theory of metric spaces, qualitative theory of differential/difference equations and dynamical systems.
The equations used to describe dynamic properties of physical systems are often nonlinear, and it is rarely possible to find their solutions. Although numerical solutions are impractical and graphical techniques are not useful for many types of systems, there are different theorems and methods that are useful regarding qualitative properties of nonlinear systems and their solutions—system stability being the most crucial property. Without stability, a system will not have value. Nonlinear Systems Stability Analysis: Lyapunov-Based Approach introduces advanced tools for stability analysis of nonlinear systems. It presents the most recent progress in stability analysis and provides a complete review of the dynamic systems stability analysis methods using Lyapunov approaches. The author discusses standard stability techniques, highlighting their shortcomings, and also describes recent developments in stability analysis that can improve applicability of the standard methods. The text covers mostly new topics such as stability of homogonous nonlinear systems and higher order Lyapunov functions derivatives for stability analysis. It also addresses special classes of nonlinear systems including time-delayed and fuzzy systems. Presenting new methods, this book provides a nearly complete set of methods for constructing Lyapunov functions in both autonomous and nonautonomous systems, touching on new topics that open up novel research possibilities. Gathering a body of research into one volume, this text offers information to help engineers design stable systems using practice-oriented methods and can be used for graduate courses in a range of engineering disciplines.
The theory of nonautonomous dynamical systems in both of its formulations as processes and skew product flows is developed systematically in this book. The focus is on dissipative systems and nonautonomous attractors, in particular the recently introduced concept of pullback attractors. Linearization theory, invariant manifolds, Lyapunov functions, Morse decompositions and bifurcations for nonautonomous systems and set-valued generalizations are also considered as well as applications to numerical approximations, switching systems and synchronization. Parallels with corresponding theories of control and random dynamical systems are briefly sketched. With its clear and systematic exposition, many examples and exercises, as well as its interesting applications, this book can serve as a text at the beginning graduate level. It is also useful for those who wish to begin their own independent research in this rapidly developing area.
The first systematic presentation of the theory of dynamical systems under the influence of randomness, this book includes products of random mappings as well as random and stochastic differential equations. The basic multiplicative ergodic theorem is presented, providing a random substitute for linear algebra. On its basis, many applications are detailed. Numerous instructive examples are treated analytically or numerically.
A systematic introduction to the core of smooth ergodic theory. An expanded version of an earlier work by the same authors, it describes the general (abstract) theory of Lyapunov exponents and the theory's applications to the stability theory of differential equations, the stable manifold theory, absolute continuity of stable manifolds, and the ergodic theory of dynamical systems with nonzero Lyapunov exponents (including geodesic flows). It could be used as a primary text for a course on nonuniform hyperbolic theory or as supplemental reading for a course on dynamical systems. Assumes a basic knowledge of real analysis, measure theory, differential equations, and topology. c. Book News Inc.
Mathematics in Science and Engineering, Volume 30: Stability of Motion deals with the problem of stability of motion. This volume investigates the problem of stability of the unperturbed motion in cases such as the system of differential equations for the perturbed motion is autonomie and the characteristic equation of the linear system that gives the first approximation has a double zero root. When the order of the system is larger than two (n > 2), all the remaining roots have negative real parts. The double root corresponds to a multiple elementary divisor of the characteristic matrix. This book is a good reference for mathematicians, students, and specialists conducting work on the stability of motion.
This volume contains the notes from five lecture courses devoted to nonautonomous differential systems, in which appropriate topological and dynamical techniques were described and applied to a variety of problems. The courses took place during the C.I.M.E. Session "Stability and Bifurcation Problems for Non-Autonomous Differential Equations," held in Cetraro, Italy, June 19-25 2011. Anna Capietto and Jean Mawhin lectured on nonlinear boundary value problems; they applied the Maslov index and degree-theoretic methods in this context. Rafael Ortega discussed the theory of twist maps with nonperiodic phase and presented applications. Peter Kloeden and Sylvia Novo showed how dynamical methods can be used to study the stability/bifurcation properties of bounded solutions and of attracting sets for nonautonomous differential and functional-differential equations. The volume will be of interest to all researchers working in these and related fields.
