  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.
