Contemporary Mathematics 1989; 357 pp; softcover Volume: 99 ISBN10: 0821851055 ISBN13: 9780821851050 List Price: US$60 Member Price: US$48 Order Code: CONM/99
 The last few years have seen a number of major developments demonstrating that the longterm behavior of solutions of a very large class of partial differential equations possesses a striking resemblance to the behavior of solutions of finite dimensional dynamical systems, or ordinary differential equations. The first of these advances was the discovery that a dissipative PDE has a compact, global attractor with finite Hausdorff and fractal dimensions. More recently, it was shown that some of these PDEs possess a finite dimensional inertial manifoldthat is, an invariant manifold containing the attractor and exponentially attractive trajectories. With the improved understanding of the exact connection between finite dimensional dynamical systems and various classes of dissipative PDEs, it is now 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 mathematical models, including continuum flows. Surprisingly, a number of distributed systems from continuum mechanics have been found to exhibit the same nontrivial dynamic behavior as observed in lowdimensional 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 infinitedimensional systems. This book represents the proceedings of an AMSIMSSIAM Summer Research Conference, held in July, 1987 at the University of Colorado at Boulder. Bringing together mathematicians and physicists, the conference provided a forum for presentations on the latest developments in the field and fostered lively interactions on open questions and future directions. With contributions from some of the top experts, these proceedings will provide readers with an overview of this vital area of research. Table of Contents  R. Temam  Dynamical systems in infinite dimension
 P. Constantin  A construction of inertial manifolds
 M. Tabor  Analytic structure of dynamical systems
 G. Sell  Hausdorff and Lyapunov dimensions for gradient systems
 D. Armbruster  Persistent homoclinic orbits
 M. S. Jolly  Orientation of saddle connections for a reactiondiffusion equation
 C. R. Doering, J. D. Gibbon, D. D. Holm, and B. Nicolaenko  Finite dimensionality in the complex LandauGinzburg equation
 J. M. Ghidaglia and R. Temam  Periodic dynamical systems with applications to SineGordon equations: Estimates of the fractal dimension of the universal attractor
 B. Nicolaenko  Inertial manifolds for models of compressible gas dynamics
 W. W. Zachary and T. Gill  Existence and finitedimensionality of universal attractors for the LandauLifschitz equations of ferromagnetism
 M. I. Weinstein  The nonlinear Schrödinger equationsingularity formation, stability and dispersion
 A. Mazer and T. Ratiu  Formal stability of twodimensional selfgravitating rotating disks
 E. Van der Groesen  A deterministic approach towards selforganization in continuous media
 L. Sirovich  Low dimensional description of complicated phenomena
 E. Kostelich and J. Yorke  Using dynamic embedding methods to analyze experimental data
 I. G. Kevrekidis and R. Ecke  Global bifurcation in maps of the Plane and RayleighBernard convection
 E. Knobloch, A. Deane, and J. Toomre  A model of doublediffusive convection with periodic boundary conditions
 K. Gustafson, K. Halasi, and R. Leben  Controversies concerning finite/infinite sequences of fluid corner vortices
