Determining functionals and finite-dimensional reduction for dissipative PDEs revisited

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