# The Connection between Infinite Dimensional and Finite Dimensional Dynamical Systems

Share this page *Edited by *
*Basil Nicolaenko*

The last few years have seen a number of major developments demonstrating that the long-term 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 manifold–that 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 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.

This book represents the proceedings of an AMS-IMS-SIAM 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

## The Connection between Infinite Dimensional and Finite Dimensional Dynamical Systems

- Contents ix10 free
- Preface xi12 free
- Dynamical systems in infinite dimension 114 free
- A construction of inertial manifolds 2740
- Analytic structure of dynamical systems 6376
- Hausdorff and Lyapunov dimensions for gradient systems 8598
- Persistent heteroclinic orbits 93106
- Orientation of saddle connections for a reaction diffusion equation 105118
- Finite dimensionality in the complex Ginzburg-Landau equation 117130
- Periodic dynamical system with application to Sine-Gordon equations: Estimates on the fractal dimension of the universal attractor 143156
- Inertial manifolds for models of compressible gas dynamics 165178
- Existence and finite-dimensionality of universal attractors for the Landau-Lifschitzequations of ferromagnetism 181194
- The nonlinear Schrodinger equation—singularity formation, stability and dispersion 213226
- Formal stability of two-dimensional self-gravitating rotating disks 233246
- A deterministic approach towards self-organization in continuous media 259272
- Low dimensional description of complicated phenomena 277290
- Using dynamic embedding methods to analyze experimental data 307326
- Global bifurcations in maps of the plane and in Rayleigh-Bénard convection 313332
- A model of double-diffusive convection with periodic boundary conditions 339358
- Controversies concerning finite/infinite sequences of fluid corner vortices 351370