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## Über dieses Buch

Where do solutions go, and how do they behave en route? These are two of the major questions addressed by the qualita­ tive theory of differential equations. The purpose of this book is to answer these questions for certain classes of equa­ tions by recourse to the framework of semidynamical systems (or topological dynamics as it is sometimes called). This approach makes it possible to treat a seemingly broad range of equations from nonautonomous ordinary differential equa­ tions and partial differential equations to stochastic differ­ ential equations. The methods are not limited to the examples presented here, though. The basic idea is this: Embed some representation of the solutions of the equation (and perhaps the equation itself) in an appropriate function space. This space serves as the phase space for the semidynamical system. The phase map must be chosen so as to generate solutions to the equation from an initial value. In most instances it is necessary to provide a "weak" topology on the phase space. Typically the space is infinite dimensional. These considerations motivate the requirement to study semidynamical systems in non locally compact spaces. Our objective here is to present only those results needed for the kinds of applications one is likely to encounter in differen­ tial equations. Additional properties and extensions of ab­ stract semidynamical systems are left as exercises. The power of the semidynamical framework makes it possible to character- Preface ize the asymptotic behavior of the solutions of such a wide class of equations.

## Inhaltsverzeichnis

### Chapter I. Basic Definitions and Properties

Abstract
After the appropriate definitions in Section 2 we present in Section 3 a simple example of a semidynamical system in an infinite dimensional space. The example arises in conjunction with a Poisson process. The objective of Section 4 is to extend semidynamical systems to the negative time domain. This results in a classification of the maximal time domains of semidynamical systems. In Section 5 we characterize critical and periodic motions and prove an extension theorem for periodic motions. Examples are presented which distinguish semidynamical system behavior from its precursor, dynamical systems. Section 6 is devoted to both an algebraic and a topological classification of positive orbits. The chapter is closed following a reparametrization theorem. It permits us to treat global semidynamical systems throughout the book. This results in a considerable savings as regards the theory without losing the applicability of the results to many important examples.
Stephen H. Saperstone

### Chapter II. Invariance, Limit Sets, and Stability

Abstract
The main concerns of this chapter are two-fold. Firstly, where do positive motions go as t → ∞, and secondly, what can we say about the behavior of the resulting limiting orbits?
Stephen H. Saperstone

### Chapter III. Motions in Metric Space

Abstract
Stability and attraction for sets as defined in Chapter II only yield information on the behavior of motions near a positively invariant set. Much detail is obscured by only considering the positive orbit or its hull. As Sell points out in [1], stability and attraction of sets are too crude to give much information about the behavior of motions within a set. A finer tool is needed here. In part, this can be accomplished by endowing the phase space with a metric or uniform structure. Also, the consideration of almost periodic motions requires completeness of the phase space. This added structure allows us to answer the question posed in the opening paragraph of Section 4 of Chapter II; namely, when does a semidynamical system extend (uniquely) to a dynamical system? Moreover, in this setting we can complete- the classification of compact positively minimal sets into the closure of recurrent, uniformly recurrent, almost periodic, periodic, and critical motions.
Stephen H. Saperstone

### Chapter IV. Nonautonomous Ordinary Differential Equations

Abstract
The solutions of the autonomous ordinary differential equation
$$\dot{x} = f(x)$$
(1.1)
(where ẋ stands for dx/dt) give rise to a semidynamical (even dynamical) system on IRd provided f: W → IRd is continuous on the open subset W ⊂ IRd and the solutions of Equation (1.1) through any point (x0,t0) ∈ W × IR are uniquely defined and remain in W for all time. In fact, if Φ(x0;t) denotes the solution of Equation (1.1) through (x0,0) evaluated at time t ∈ IR+, it can be verified that (W,Φ) is a semidynamical system.
Stephen H. Saperstone

### Chapter V. Semidynamical Systems in Banach Space

Abstract
Material fundamental to the existence and qualitative behavior of partial differential equations and differential delay equations (to name just two areas) are developed in this chapter. The general formulation is an evolution equation in a Banach space. The work of Crandall and Liggett on the nonlinear version of the Hille — Yosida — Phillips theorem for linear semigroups has spawned an elegant analysis of nonlinear evolution equations in Banach spaces. One special feature of the semigroup generation theorem (linear or nonlinear) is that we obtain a representation of the solutions to du/dt + Au = 0 in terms of the operator A. The classical approach was to establish the existence, uniqueness, and continuous dependence of the solutions of the particular partial differential equation, for example, and then demonstrate that the solutions generate a semigroup. This was essentially the approach we also took in Chapter IV.
Stephen H. Saperstone

### Chapter VI. Functional Differential Equations

Abstract
A distinguishing feature of ordinary differential equations is that the future behavior of solutions depends only upon the present (initial) values of the solution. Numerous physical, economic, biological, and social systems, though, exhibit hereditary dependence. That is, the future state of the system depends not only upon the present state, but also upon past states. Models of such systems must take into account this hereditary effect. We illustrate this with some examples.
Stephen H. Saperstone

### Chapter VII. Stochastic Dynamical Systems

Abstract
We turn to a semidynamical system which is generated by a Markov process. Here again we obtain, in general, a non-differentiable system.
Stephen H. Saperstone

### Chapter VIII. Weak Semidynamical Systems and Processes

Abstract
Many of the important properties of semidynamical systems which were developed in the first three chapters can essentially be obtained with a weaker continuity axiom (Definition 2.1(iii) of Chapter I); namely, assume that π(x,t) is only continuous in x ∈ X. In particular, we still obtain weak in-variance of compact positive limit sets. In addition, if the continuity in x is uniform with respect to t ∈ IR+, then (X,π) extends to a weak dynamical system on the positive limit sets. Moreover, the positive limit sets will then be minimal with respect to this flow. Finally, we will still be able to show, as in Chapter III, that the positive limit sets are equi-almost periodic.
Stephen H. Saperstone

### Backmatter

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