To treat systems in which there is time dependence in the dynamics, due either to variation of an external parameter or to evolution of the system itself, the scheme for branching paths must be devised so as to achieve sampling which is distributed evenly in time and which takes account of changing fluxes through different regions of the phase space.
The process of branching requires that identical paths can be made to diverge from each other, such as by changing the seed in the computer's random number generator.
For systems which would be naturally considered as deterministic, it may be possible to inject an element of randomness, for instance by coupling to a fluctuating heat bath or by adding random perturbations to account for some elements of the simulation which are not modelled explicitly but which exist in the real system.
The amount of over or under-sampling (the branching density) is decided based on some system-specific 'progress coordinate' which measures progress toward a rare event of interest.
The probability of selecting a configuration as the starting point for a new path segment is conditioned jointly by its probability of appearing in an unbiased simulation and by the local flux forwards in the progress coordinate, with a small flux leading adaptively to a larger oversampling.