  The presence of large event-by-event flow fluctuations in heavy ion
collisions at RHIC and the LHC provides an opportunity to study a broad class
of flow observables. This paper explores the correlations among harmonic flow
coefficients $v_n$ and their phases $\Phi_n$, and the rapidity fluctuation of
$v_n$. The study is carried out usin Pb+Pb events generated by the AMPT model
with fixed impact parameter. The overall ellipticity/triangularity of events is
varied by selecting on the eccentricities $\epsilon_n$ or the magnitudes of the
flow vector $q_n$ for n=2 and 3, respectively. The responses of the $v_n$, the
event-plane correlations, and the rapidity fluctuations, to the change in
$\epsilon_n$ and $q_n$ are then systematized. Strong positive correlations are
observed among all even harmonics $v_2, v_4$, and $v_6$ (all increase with
$q_2$), between $v_2$ and $v_5$ (both increase with $q_2$) and between $v_3$
and $v_5$ (both increase with $q_3$), consistent with the effects of nonlinear
collective response. In contrast, an anti-correlation is observed between $v_2$
and $v_3$ similar to that seen between $\epsilon_2$ and $\epsilon_3$. These
correlation patterns are found to be independent of whether selecting on $q_n$
or $\epsilon_n$, validating the ability of $q_n$ in selecting the initial
geometry. A forward/backward asymmetry of $v_n(\eta)$ is observed for events
selected on $q_n$ but not on $\epsilon_n$, reflecting dynamical fluctuations
exposed by the $q_n$ selection. Many event-plane correlators show good
agreement between $q_n$ and $\epsilon_n$ selections, suggesting that their
variations with $q_n$ are controlled by the change of $\epsilon_n$ in the
initial geometry. Hence these correlators may serve as promising observables
for disentangling the fluctuations generated in various stages of the evolution
of the matter created in heavy ion collisions.
