  This paper concerns stochastic perturbations of piecewise-smooth ODE systems
relevant for vibro-impacting dynamics, where impact events constitute the
primary source of randomness. Such systems are characterised by the existence
of switching manifolds that divide the phase space into regions where the
system is smooth. The initiation of impacts is captured by a grazing
bifurcation, at which a periodic orbit describing motion without impacts
develops a tangential intersection with a switching manifold. Oscillatory
dynamics near regular grazing bifurcations are described by piecewise-smooth
maps involving a square-root singularity, known as Nordmark maps. We consider
three scenarios where coloured noise only affects impacting dynamics, and
derive three two-dimensional stochastic Nordmark maps with the noise appearing
in different nonlinear or multiplicative ways, depending on the source of the
noise. Consequently the stochastic dynamics differs between the three noise
sources, and is fundamentally different to that of a Nordmark map with additive
noise. This critical dependence on the nature of the noise is illustrated with
a prototypical one-degree-of-freedom impact oscillator.

Dynamical Systems