NDLI: Stochastic Regular Grazing Bifurcations

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Simpson, D. J. W. Hogan, S. J. Kuske, R.
Copyright Year	2013
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 Poincaré 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 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 $\sqrt{\varepsilon}$, 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.
Starting Page	533
Ending Page	559
Page Count	27
File Format	PDF
DOI	10.1137/120884286
e-ISSN	15360040
Issue Number	2
Volume Number	12
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2013-04-04
Access Restriction	Subscribed
Subject Keyword	Brownian motion vibro-impacting system piecewise-smooth noise Bifurcation Bifurcations of limit cycles and periodic orbits stochastic perturbation
Content Type	Text
Resource Type	Article

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