  The determining modes for the two-dimensional incompressible Navier-Stokes
equations (NSE) are shown to satisfy an ordinary differential equation of the
form $dv/dt=F(v)$, in the Banach space, $X$, of all bounded continuous
functions of the variable $s\in\mathbb{R}$ with values in certain
finite-dimensional linear space. This new evolution ODE, named {\it determining
form}, induces an infinite-dimensional dynamical system in the space $X$ which
is noteworthy for two reasons. One is that $F$ is globally Lipschitz from $X$
into itself. The other is that the long-term dynamics of the determining form
contains that of the NSE; the traveling wave solutions of the determining form,
i.e., those of the form $v(t,s)=v_0(t+s)$, correspond exactly to initial data
$v_0$ that are projections of solutions of the global attractor of the NSE onto
the determining modes. The determining form is also shown to be dissipative; an
estimate for the radius of an absorbing ball is derived in terms of the number
of determining modes and the Grashof number (a dimensionless physical
parameter). Finally, a unified approach is outlined for an ODE satisfied by a
variety of other determining parameters such as nodal values, finite volumes,
and finite elements.
