Introduction

Abstract

Recently, considerable theoretical and computational evidence has accumulated supporting the remarkable similarities between the long-time evolution of solutions of dissipative partial differential equations (PDEs) and solutions of finite-dimensional dynamical systems, or ordinary differential equations (ODEs). For the latter, numerous studies have discovered and analyzed complex dynamical bifurcations of finite vector fields [CoE, De, Sch, GH, MeP, ChH, BPV]. Computer simulations for the dynamics of many dissipative PDEs evidence an equally rich complexity [HN1, HN2, HNZ, BLMcLO]. The connection between the long-time behavior of finite differential systems and that of PDEs was first established by the discovery that dissipative PDEs possess a finite number of asymptotic degrees of freedom: they have a compact, universal attractor X with finite Hausdorff and fractal dimension (modulo some regularity conditions) [BV, BV1, CF1, CFT, DO, He, HI, HMO, MeP, MP, NST, NST1, T]. Estimates on the number of such degrees of freedom have been obtained for two- and three-dimensional turbulent continuum flows [CF1, CFT, CFMT]. Still, such results do not imply that, for a given dissipative PDE, the asymptotic behavior and in particular the universal attractor X coincide with those of an appropriate differential equation. Recently, it has been shown that for certain dissipative PDEs this is indeed the case.