  Dense stellar clusters are natural sites for the origin and evolution of
exotic objects such as relativistic binaries (potential gravitational wave
sources), blue stragglers, etc. We investigate the secular dynamics of a binary
system driven by the global tidal field of an axisymmetric stellar cluster in
which the binary orbits. In a companion paper (Hamilton & Rafikov 2019a) we
developed a general Hamiltonian framework describing such systems. The
effective (doubly-averaged) Hamiltonian derived there encapsulates all
information about the tidal potential experienced by the binary in its orbit
around the cluster in a single parameter $\Gamma$. Here we provide a thorough
exploration of the phase-space of the corresponding secular problem as $\Gamma$
is varied. We find that for $\Gamma > 1/5$ the phase-space structure and the
evolution of binary orbital element are qualitatively similar to the
Lidov-Kozai problem. However, this is only one of four possible regimes,
because the dynamics are qualitatively changed by bifurcations at $\Gamma =
1/5,0,-1/5$. We show how the dynamics are altered in each regime and calculate
characteristics such as secular evolution timescale, maximum possible
eccentricity, etc. We verify the predictions of our doubly-averaged formalism
numerically and find it to be very accurate when its underlying assumptions are
fulfilled, typically meaning that the secular timescale should exceed the
period of the binary around the cluster by $\gtrsim 10-10^2$ (depending on the
cluster potential and binary orbit). Our results may be relevant for
understanding the nature of a variety of exotic systems harboured by stellar
clusters.
