Nonautonomous Dynamics

This book emphasizes those topological methods (of dynamical systems) and theories that are useful in the study of different classes of nonautonomous evolutionary equations. The content is developed over six chapters, providing a thorough introduction to the techniques used in the Chapters III-VI described by Chapter I-II. The author gives a systematic treatment of the basic mathematical theory and constructive methods for Nonautonomous Dynamics. They show how these diverse topics are connected to other important parts of mathematics, including Topology, Functional Analysis and Qualitative Theory of Differential/Difference Equations. Throughout the book a nice balance is maintained between rigorous mathematics and applications (ordinary differential/difference equations, functional differential equations and partial difference equations). The primary readership includes graduate and PhD students and researchers in in the field of dynamical systems and their applications (control theory, economic dynamics, mathematical theory of climate, population dynamics, oscillation theory etc).

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.

Nonautonomous dynamics describes the qualitative behavior of evolutionary differential and difference equations, whose right-hand side is explicitly time dependent. Over recent years, the theory of such systems has developed into a highly active field related to, yet recognizably distinct from that of classical autonomous dynamical systems. This development was motivated by problems of applied mathematics, in particular in the life sciences where genuinely nonautonomous systems abound. The purpose of this monograph is to indicate through selected, representative examples how often nonautonomous systems occur in the life sciences and to outline the new concepts and tools from the theory of nonautonomous dynamical systems that are now available for their investigation.

The nature of time in a nonautonomous dynamical system is very different from that in autonomous systems, which depend only on the time that has elapsed since starting rather than on the actual time itself. Consequently, limiting objects may not exist in actual time as in autonomous systems. New concepts of attractors in nonautonomous dynamical system are thus required.In addition, the definition of a dynamical system itself needs to be generalised to the nonautonomous context. Here two possibilities are considered: two-parameter semigroups or processes and the skew product flows. Their attractors are defined in terms of families of sets that are mapped onto each other under the dynamics rather than a single set as in autonomous systems. Two types of attraction are now possible: pullback attraction, which depends on the behaviour from the system in the distant past, and forward attraction, which depends on the behaviour of the system in the distant future. These are generally independent of each other.The component subsets of pullback and forward attractors exist in actual time. The asymptotic behaviour in the future limit is characterised by omega-limit sets, in terms of which form what are called forward attracting sets. They are generally not invariant in the conventional sense, but are asymptotically invariant in general and, if the future dynamics is appropriately uniform, also asymptotically negatively invariant.Much of this book is based on lectures given by the authors in Frankfurt and Wuhan. It was written mainly when the first author held a 'Thousand Expert' Professorship at the Huazhong University of Science and Technology in Wuhan.

The goal of this book is to provide an approach to the corresponding geometric theory of nonautonomous discrete dynamical systems in infinite-dimensional spaces by virtue of 2-parameter semigroups (processes).

This book offers an introduction to the theory of non-autonomous and stochastic dynamical systems, with a focus on the importance of the theory in the Applied Sciences. It starts by discussing the basic concepts from the theory of autonomous dynamical systems, which are easier to understand and can be used as the motivation for the non-autonomous and stochastic situations. The book subsequently establishes a framework for non-autonomous dynamical systems, and in particular describes the various approaches currently available for analysing the long-term behaviour of non-autonomous problems. Here, the major focus is on the novel theory of pullback attractors, which is still under development. In turn, the third part represents the main body of the book, introducing the theory of random dynamical systems and random attractors and revealing how it may be a suitable candidate for handling realistic models with stochasticity. A discussion of future research directions serves to round out the coverage.

The study of attractors of dynamical systems occupies an important position in the modern qualitative theory of differential equations. This engaging volume presents an authoritative overview of both autonomous and non-autonomous dynamical systems, including the global compact attractor. From an in-depth introduction to the different types of dissipativity and attraction, the book takes a comprehensive look at the connections between them, and critically discusses applications of general results to different classes of differential equations. Intended for experts in qualitative theory of differential equations, dynamical systems and their applications, this accessible book can also serve as an important resource for senior students and lecturers.

Although, bifurcation theory of equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. In this book, two different approaches are developed which are based on special definitions of local attractivity and repulsivity. It is shown that these notions lead to nonautonomous Morse decompositions.