Determining functionals and finite-dimensional reduction for dissipative PDEs revisited

TL;DR

This paper revisits the concept of determining functionals for dissipative PDEs, showing that their minimal number is closely related to the equilibrium set's dimension rather than the attractor's complexity, with implications for system analysis.

Contribution

It establishes that the determining dimension is primarily linked to the equilibrium set's dimension, challenging the traditional focus on attractor complexity.

Findings

Determining dimension equals one when the equilibrium set is finite.

Optimal number of functionals relates to equilibrium set, not attractor complexity.

Results supported by explicit examples.

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.