  We establish the dynamical attractor behavior in scalar-tensor theories of
dark energy, providing a powerful framework to analyze classes of theories,
predicting common evolutionary characteristics that can be compared against
cosmological constraints. In the Jordan frame the theories are viewed as a
coupling between a scalar field, \Phi, and the Ricci scalar, R, F(\Phi)R. The
Jordan frame evolution is described in terms of dynamical variables m \equiv
d\ln F/d\ln \Phi and r \equiv -\Phi F/f, where F(\Phi) = d f(\Phi)/d\Phi. The
evolution can be alternatively viewed in the Einstein frame as a general
coupling between scalar dark energy and matter, \beta. We present a complete,
consistent picture of evolution in the Einstein and Jordan frames and consider
the conditions on the form of the coupling F and \beta required to give the
observed cold dark matter (CDM) dominated era that transitions into a late time
accelerative phase, including transitory accelerative eras that have not
previously been investigated. We find five classes of evolutionary behavior of
which four are qualitatively similar to those for f(R) theories (which have
\beta=1/2). The fifth class exists only for |\beta| < \sqrt{3}/4, i.e. not for
f(R) theories. In models giving transitory late time acceleration, we find a
viable accelerative region of the (r,m) plane accessible to scalar-tensor
theories with any coupling, \beta (at least in the range |\beta| \leq 1/2,
which we study in detail), and an additional region open only to theories with
|\beta| < \sqrt{3}/4.
