Stochastic Regular Grazing Bifurcations

A grazing bifurcation corresponds to the collision of a periodic orbit with a switching manifold in a piecewise-smooth ODE system and often generates complicated dynamics. The lowest order terms of the induced Poincare map expanded about a regular grazing bifurcation constitute a Nordmark map. In this paper we study a normal form of the Nordmark map in two dimensions with additive Gaussian noise of amplitude $\varepsilon$. We show that this particular noise formulation arises in a general setting and consider a harmonically forced linear oscillator subject to compliant impacts to illustrate the accuracy of the map. Numerically computed invariant densities of the stochastic Nordmark map can take highly irregular forms or, if there exists an attracting period-$n$ solution when $\varepsilon=0$, be well approximated by the sum of $n$ Gaussian densities centered about each point of the deterministic solution, and scaled by $\frac{1}{n}$, for sufficiently small $\varepsilon>0$. We explain the irregular forms an...