In the nonconvex case solutions of rate-independent systems may develop jumps
as a function of time. To model such jumps, we adopt the philosophy that rate
independence should be considered as limit of systems with smaller and smaller
viscosity. For the finite-dimensional case we study the vanishing-viscosity
limit of doubly nonlinear equations given in terms of a differentiable energy
functional and a dissipation potential which is a viscous regularization of a
given rate-independent dissipation potential. The resulting definition of 'BV
solutions' involves, in a nontrivial way, both the rate-independent and the
viscous dissipation potential, which play a crucial role in the description of
the associated jump trajectories. We shall prove a general convergence result
for the time-continuous and for the time-discretized viscous approximations and
establish various properties of the limiting BV solutions. In particular, we
shall provide a careful description of the jumps and compare the new notion of
solutions with the related concepts of energetic and local solutions to
rate-independent systems.