  This article studies the interrelation between the determining modes property
in the two-dimensional (2D) Navier-Stokes equations (NSE) of incompressible
fluids and the synchronization property of two filtering algorithms for
continuous data assimilation applied to the 2D NSE. These two properties are
realized as manifestations of a more general phenomenon of "self-synchronous
intertwinement". It is shown that this concept is a logically stronger form of
asymptotic enslavement, as characterized by the existence of finitely many
determining modes in the 2D NSE. In particular, this stronger form is shown to
imply convergence of the synchronization filter and the nudging filter from
continuous data assimilation (CDA), and then subsequently invoked to show that
convergence in these filters implies that the 2D NSE possesses finitely many
determining modes. The main achievement of this article is to therefore
identify a new concept, that of self-synchronous intertwinement, through which
a rigorous relationship between the determining modes property and
synchronization in these CDA filters is established and made decisively clear.
The theoretical results are then complemented by numerical experiments that
confirm the conclusions of the theorems.
