Stochastic Regular Grazing Bifurcations

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, ε. 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 ε = 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 ε > 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 √ ε, 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.

Metastability for discontinuous dynamical systems under Lévy noise: Case study on Amazonian Vegetation

“…This classification allows for the description of the behaviour of solutions in the neighbourhood of isolated singular points and detects the types of points that disturb the uniqueness of solutions. Stochastic bifurcation analysis has also been studied for different types of models: for smooth and discontinuous oscillators 34 , and for piecewise-smooth ODEs in two dimensions with additive Gaussian noise 35 , among others. The dynamical behaviour of stochastically perturbed solutions near the switching manifold has been studied in refs 36 and 37 .…”

The influence of localized randomness on regular grazing bifurcations with applications to impacting dynamics

“…To motivate and illustrate our results for stochastic Nordmark maps, we consider the prototypical vibro-impacting system shown in Fig. 4 and studied in [37,40,41]. This system consists of a harmonically forced one-degree-of-freedom linear oscillator that experiences compliant (or soft) impacts with a support, and we use the following non-dimensionalised equations to model the dynamics:ü…”

“…He found that noise blurs bifurcation diagrams and washes out high-period solutions in the same manner as for smooth maps, such as the logistic map (Mayer-Kress and Haken, 1981; Crutchfield et al., 1982). Recently it was shown that white noise added to (1) translates to additive white noise in (3) (Simpson et al., 2013). Such a noise formulation may be sensible for vibro-impacting systems for which a forcing term or external fluctuations represent a significant source of uncertainty.…”

The behavior of bilinear impact oscillators subjected to random forcings

“…This explanation for why such a flipping behavior is observed can be provided with the help of bifurcation diagrams constructed in the presence of noise. Such bifurcation diagrams constructed give an idea of the spread of the modulated periodic attractor [11].…”