  We investigate a local incremental stationary scheme for the numerical
solution of rate-independent systems. Such systems are characterized by a
(possibly) non-convex energy and a dissipation potential, which is positively
homogeneous of degree one. Due to the non-convexity of the energy, the system
does in general not admit a time-continuous solution. In order to resolve these
potential discontinuities, the algorithm produces a sequence of state variables
and physical time points as functions of a curve parameter. The main novelty of
our approach in comparison to existing methods is an adaptive choice of the
step size for the update of the curve parameter depending on a prescribed
tolerance for the residua in the energy-dissipation balance and in a
complementarity relation concerning the so-called local stability condition. It
is proven that, for tolerance tending to zero, the piecewise affine
approximations generated by the algorithm converge (weakly) to a so-called
$\mathbb{V}$-parametrized balanced viscosity solution. Numerical experiments
illustrate the theoretical findings and show that an adaptive choice of the
step size indeed pays off as they lead to a significant increase of the step
size during sticking and in viscous jumps.
