Stochastic Regular Grazing Bifurcations

TL;DR

This paper analyzes how stochastic effects influence grazing bifurcations in piecewise-smooth systems, using a normal form of the Nordmark map with Gaussian noise to understand invariant densities and transition sequences.

Contribution

It introduces a stochastic normal form of the Nordmark map with Gaussian noise and explores its invariant densities and bifurcation transitions.

Findings

Invariant densities can be highly irregular or Gaussian-like.

Density size scales with the square root of noise amplitude near bifurcation.

Transitions between dynamical regimes are characterized as parameters vary.

Abstract

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, epsilson [e]. 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 e = 0, be well approximated by the sum of n Gaussian densities centred about each point of the deterministic solution, and scaled by 1/n, for sufficiently small e > 0. We explain the irregular forms and calculate the covariance matrices associated with the Gaussian approximations in terms of the parameters of the map. Close to the grazing bifurcation the size of the invariant density may be proportional to the square-root of e, as a consequence of a square-root singularity in the map. Sequences of transitions between different dynamical regimes that occur as the primary bifurcation parameter is varied have not been described previously.