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
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