局部随机性对规则擦边分岔的影响及其在碰撞动力学中的应用
The influence of localised randomness on regular grazing bifurcations with applications to impacting dynamics
摘要：
本文研究了与碰撞振动动力学相关的分段光滑常微分方程系统的随机扰动，其中碰撞事件构成了随机性的主要来源。这种系统的特征在于存在切换流形，切换流形将相空间划分为系统光滑的区域。碰撞的起始被掠食分叉捕获，在该分叉处，描述没有碰撞的运动的周期性轨道与切换歧管形成切向相交。正则掠食分岔附近的振荡动力学是由包含平方根奇异性的分段光滑映射描述的，称为Nordmark映射。我们考虑三种情况下，有色噪声只影响影响动力学，并推导出三个二维随机Nordmark地图的噪声出现在不同的非线性或乘法的方式，这取决于噪声的来源。因此，随机动力学不同的三个噪声源之间，是根本不同的Nordmark地图与加性噪声。这种关键的依赖性的噪声的性质说明了一个原型的一个自由度的影响振荡器。
英文摘要：
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.
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