The 1986 NATO Advanced Study Insti tute on Dynamics of Infini te Dimensional Systems was held at the Instituto Superior Tecnico. Lisbon. Portugal. In recent years. there have been several research workers who have been considering partial differential equations and functional differential equations as dynamical systems on function spaces. Such approaches have led to the formulation of more theoretical problems that need to be investigated. In the applications. the theoretical ideas have contributed significantly to a better understanding of phenomena that have been experimentally and computationally observed. The investigators of this development come wi th several different backgrounds - some from classical partial differential equations. some from classical ordinary differential equations and some interested in specific applications. Each group has special ideas and often these ideas have not been transmitted from one group to another. The purpose of this NATO Workshop was to bring together research workers from these various areas. It provided asoundboard for the impact of the ideas of each respective discipline. We believe that goal was accomplished. but time will be a better judge. We have included the list of participants at the workshop. with most of these giving a presentation. Although the proceedings do not include all of the presentations. it is a good representative sampie. We wish to express our gratitude to NATO. and.to Dr. M. di Lullo of NATO. who unfortunately did not live to see the completion of this project.

Table of Contents:
Semilinear Parabolic Systems Under Nonlinear Boundary Conditions.- The Shadowing Lemma for Elliptic PDE.- Coagulation-Fragmentation Dynamics.- Functional Differential Equations and Jensen’s Inequality.- Method of Upper and Lower Solutions for Nonlinear Integral Equations and an Application to an Infectious Disease Model.- Competition of Azimuthal Modes and Quasi-Periodic Flows in the Couette-Taylor Problem.- On Bifurcation for Variational Problems.- Nilpotent Normal Form in Dimension 4.- Perturbed Dual Semigroups and Delay Equations.- On Operators Which Leave Invariant a Half-Space.- Global Hopf Bifurcation in Reaction Diffusion Systems with Symmetry.- Longtime Behavior for a Class of Abstract Integrodifferential Equations.- Describing the Flow on the Attractor of One Dimensional Reaction Diffusion Equations by Systems of ODE.- Asymptotic Behavior of Gradient Dissipative Systems.- Generic Properties of Equilibrium Solutions by Perturbation of the Boundary.- Complex Analytical Methods in RFDE Theory.- Qualitative Behavior of the Solutions of Periodic First Order Scalar Differential Equations with Strictly Convex Coercive Nonlinearity.- The Spectrum of Invariant Sets for Dissipative Semiflows.- Approximate Solutions to Conservation Laws via Convective Parabolic Equations: Analytical and Numerical Results.- Conley’s Connection Matrix.- Existence and Non-Existence of Finite-Dimensional Globally Attracting Invariant Manifolds in Semilinear Damped Wave Equations.- SL E P Method to the Stability of Singularly Perturbed Solutions with Multiple Internal Transition Layers in Reaction-Diffusion Systems.- Iterated Nonlinear Maps and Hilbert’s Projective Metric: A Summary.- Jacobi Matrices and Transversality.- Examples of Attractors in Scalar Reaction-Diffusion Equations.-Gauge Theory of Backlund Transformations, I.- Recent Developments in the Theory of Nonlinear Scalar First and Second Order Partial Differential Equations.- Hopf Bifurcation for an Infinite Delay Functional Equation.- A Numerical Analysis of the Structure of Periodic Orbits in Autonomous Functional Differential Equations.- Oscillations and Asymptotic Behaviour for two Semilinear Hyperbolic Systems.- An Application of the Conley Index to Combustion.- Path Continuation — A Sensitivity Analysis Approach.- Confinor and Anti-confinor in Constrained “Lorenz” System.- Invariant Manifolds in Infinite Dimensions.- Linearizing Completely Integrable Systems on Complex Algebraic Tori.- On Some Dynamical Aspects of Parabolic Equations with Variable Domain.- Bifurcation from Homoclinic to Periodic Solutions by an Inclination Lemma with Pointwise Estimate.- Approximate Methods for Set Valued Differential Equations with Delays.- Bounds for the Chaotic Behavior of Newton’s Method.- List of Participants.