Determining functionals and finite-dimensional reduction for dissipative PDEs revisited

Abstract

We study the properties of linear and non-linear determining functionals for dissipative dynamical systems generated by PDEs. The main attention is payed to the lower bounds for the number of such functionals. In contradiction to the common paradigm, it is shown that the optimal number of determining functionals (the so-called determining dimension) is strongly related to the proper dimension of the set of equilibria of the considered dynamical system rather than to the dimensions of the global attractors and the complexity of the dynamics on it. In particular, in the generic case where the set of equilibria is finite, the determining dimension equals to one (in complete agreement with the Takens delayed embedding theorem) no matter how complex the underlying dynamics is. The obtained results are illustrated by a number of explicit examples.