CHAPTE R l

Introduction

In the last thirty years, many of the concepts of dynamical systems on locally

compact spaces have been adapted to infinite dimensional systems, especially

to delay differential equations and partial differential equations. These latter

systems define dynamical systems on spaces which are not locally compact. Even

when the base space is locally compact but not bounded, there is not a very

general theory of dynamical systems. In this case, one can and often does restrict

the class to one which has some type of dissipative property in order to reduce

the essential part of the flow to a compact set.

A natural concept of dissipation (which we refer to as point dissipative or

ultimately bounded) is to assume that there is a bounded set into which every

orbit eventually enters and remains. For finite dimensional spaces and motivated

by his studies of the periodically forced van der Pol equation, Levinson [1944]

was led to study this formal concept. In this case, the dynamical system was

discrete where the mapping T:

Rn

—

Rn

is the mapping taking the initial data

for the differential equation to the value of the solution at time W, where w is

the period of the forcing function. Because the space is locally compact, point

dissipative implies bounded dissipative or uniformly ultimately bounded; that is,

there is a bounded set into which the orbit of any bounded set eventually enters

and remains (see, for example, Yoshizawa [1966] or Pliss [1966]). Using this fact,

one easily sees that there is a maximal compact invariant set A such that the u-

limit set w(U) of any bounded set U belongs to A\ that is, A is a global attractor.

The u;-limit set of U is defined as w{U) = n

m

o ClU

n

m

TnU.

In particular,

the set u(U) consists of all of the limit points of the orbit of U. However, oo(U)

is generally much larger than this latter set.

In the applications of dynamical systems to infinite dimensional problems, the

base space is usually not locally compact and other ideas must come into play.

If we have a dynamical system on some Banach space X and it is known

that the orbits are precompact, then it has an u;-limit set which is compact and

invariant. At first glance, it would appear that the set B consisting of the union of

all cj-limit sets of points of the space would contain all of the relevant information

concerning the asymptotic behavior of the orbits of the flow. However, this is not

the case even in finite dimensions. The global attractor A mentioned above is not

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http://dx.doi.org/10.1090/surv/025/01