Preface
During the last few years we have seen a number of major developments which
show that the longtime behavior of solutions of a very large class of partial differential
equations (PDE's) possess a striking resemblance to the behavior of solutions of
finite dimensional dynamical systems, or ordinary differential equations (ODE's).
The first of these advances was the discovery (by a number of researchers) that a
dissipative PDE has a compact, global attractor with finite Hausdorff and fractal
dimensions. More recently it was shown that some of these PDE's possess a finite
dimensional inertial manifold, i.e., an invariant manifold that contains the attractor
and exponentially attractive trajectories. For the latter equations, the connection with
ODE's is no longer a mere analogy, instead it has become a striking reality! Indeed,
when one restricts the PDE to the inertial manifold one obtains an ODE, which we
call an inertial form for the given PDE; since an inertial manifold contains the global
attractor, it follows that the longtime behavior of solutions of a PDE with an inertial
manifold is completely determined by the inertial form.
Now that one is obtaining a better understanding of the exact connection between
finite dimensional dynamical systems and various classes of dissipative PDE's, it is
realistic to hope that the wealth of studies of such topics as bifurcations of finite
vector fields and "strange" fractal attractors can be brought to bear on various math-
ematical models including continuum flows. Surprisingly, a number of distributed
systems from continuum mechanics-as well as their infinite-dimensional models-
have been found to exhibit the same nontrivial dynamic behavior, including routes to
deterministic chaos, as observed in low-dimensional dynamical systems. As a natural
consequence of these observations, a new direction of research has arisen: detection
and analysis of finite dimensional dynamical characteristics of infinite-dimensional
systems.
The Summer Seminar on "The Connection between Infinite and Finite Dimen-
sional Dynamical Systems" was hosted by the University of Colorado at Boulder and
brought together both mathematicians and mathematical physicists.
It
succeeded as
an effective catalyst to bring forward the latest developments in the field, and fostered
lively interactions on open questions and future directions. Besides aspects of global
attractors, inertial manifolds and global bifurcations, problems of non-integrable dy-
namical systems were also discussed. A major component was the application of
these ideas to fluid dynamical systems, where practitioners have sometimes diagnosed
xi
Previous Page Next Page