  This paper investigates first-order variable metric backward forward
dynamical systems associated with monotone inclusion and convex minimization
problems in real Hilbert space. The operators are chosen so that the
backward-forward dynamical system is closely related to the forward-backward
dynamical system and has the same computational complexity. We show existence,
uniqueness, and weak asymptotic convergence of the generated trajectories and
strong convergence if one of the operators is uniformly monotone. We also
establish that an equilibrium point of the trajectory is globally exponentially
stable and monotone attractor. As a particular case, we explore similar
perspectives of the trajectories generated by a dynamical system related to the
minimization of the sum of a nonsmooth convex and a smooth convex function.
Numerical examples are given to illustrate the convergence of trajectories.
