  Finding efficient descriptions of how an environment affects a collection of
discrete quantum systems would lead to new insights into many areas of modern
physics. Markovian, or time-local, methods work well for individual systems,
but for groups a question arises: does system-bath or inter-system coupling
dominate the dissipative dynamics? The answer has profound consequences for the
long-time quantum correlations within the system. We consider two bosonic modes
coupled to a bath. By comparing an exact solution to different Markovian master
equations, we find that a smooth crossover of the equations-of-motion between
dominant inter-system and system-bath coupling exists -- but requires a
non-secular master equation. We predict a singular behaviour of the dynamics,
and show that the ultimate failure of non-secular equations of motion is
essentially a failure of the Markov approximation. Our findings justify the use
of time-local theories throughout the crossover between system-bath dominated
and inter-system-coupling dominated dynamics.
