  A one-parameter family of time-reversible systems on $\mathbb{T}^3$ is
considered. It is shown that the dynamics is not conservative, namely the
attractor and repeller intersect but not coincide. We explain this as the
manifestation of the so-called mixed dynamics phenomenon which corresponds to a
persistent intersection of the closure of stable periodic orbits and the
closure of the completely unstable periodic orbits. We search for the stable
and unstable periodic orbits indirectly, by finding non-conservative saddle
periodic orbits and heteroclinic connections between them. In this way, we are
able to claim the existence of mixed dynamics for a large range of parameter
values. We investigate local and global bifurcations that can be used for the
detection of mixed dynamics.
