SIAM Review, Volume 62, Issue 4, Page 869-897, January 2020.
In mechanics, one often describes microscopic processes such as those leading to friction between relative interfaces using macroscopic variables (relative velocity, temperature, etc.) in order to avoid models of intangible complexity. As a consequence, such macroscopic models are frequently nonsmooth, a prominent example being the Coulomb law of friction. In many cases, these models are perfectly adequate for engineering purposes. Formally, however, since the Fundamental Theorem of Existence and Uniqueness does not apply to these situations, one generally expects that these models possess forward nonuniqueness of solutions. Consequently, numerical computations of such systems might possibly unknowingly discard certain solutions. In this paper, we try to shed further light on this issue by studying solutions of a simple friction oscillator subject to stiction friction. The stiction law is a simple nonsmooth model of friction that is a modification of Coulomb based on the fundamental observation that the dynamic friction force, when the mass is in motion, is smaller than the static friction force during stick. The resulting piecewise smooth vector field of this discontinuous model does not follow the classical Filippov convention, and the concept of a Filippov solution cannot be used. Furthermore, some Carathéodory solutions, i.e., absolutely continuous solutions satisfying the differential equation in a weaker sense, are nonphysical. Therefore, we introduce the concept of stiction solutions. These are the Carathéodory solutions that are physically relevant, i.e., the ones that follow the stiction law. However, we find that some of the stiction solutions are forward nonunique in subregions of the slip onset. We call these solutions singular, in contrast to the regular stiction solutions that are forward unique. In order to further understanding of the nonunique dynamics, we then introduce a general regularization of the model. This gives a singularly perturbed problem that captures the main features of the original discontinuous problem. Using geometric singular perturbation theory, we identify a repelling slow manifold that separates the forward slipping from the forward sticking solutions, leading to high sensitivity to the initial conditions. On this slow manifold we find canard trajectories that have the physical interpretation of delaying the slip onset. Most interestingly, we find that these new solutions do not correspond to stiction solutions in the piecewise-smooth limit, and are therefore seemingly nonphysical, yet they are robust and appear generically in the class of regularizations we consider. Finally, we show that the regularized problem has a family of periodic orbits interacting with the canards. We observe that this family has a saddle stability and that it connects, in the rigid body limit, the two regular, slip-stick branches of the discontinuous problem, which are otherwise disconnected.

中文翻译：

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带有油画的静力振子：关于分段光滑非唯一性及其几何奇摄动理论的正则化解析

SIAM评论，第62卷，第4期，第869-897页，2020年1月。
在力学中，人们经常描述微观过程，例如使用宏观变量（相对速度，温度等）导致相对界面之间摩擦的过程，以避免模型无形的复杂性。结果，这种宏观模型通常是不光滑的，一个突出的例子是库仑摩擦定律。在许多情况下，这些模型完全可以满足工程目的。但是，从形式上讲，由于存在和唯一性基本定理不适用于这些情况，因此人们通常希望这些模型具有解的前向非唯一性。因此，此类系统的数值计算可能会在不知不觉中放弃某些解决方案。在本文中，我们试图通过研究受静摩擦力作用的简单摩擦振荡器的解决方案来进一步阐明这一问题。静摩擦定律是一种简单的非光滑摩擦模型，是对库仑的一种修改，它基于以下基本观察结果：当质量运动时，动态摩擦力小于粘住过程中的静态摩擦力。此不连续模型的最终分段光滑向量场不遵循经典的Filippov约定，因此无法使用Filippov解决方案的概念。此外，某些Carathéodory解，即在较弱的意义上满足微分方程的绝对连续解，是非物理的。因此，我们介绍了静摩擦解决方案的概念。这些是在物理上相关的Carathéodory解决方案，即 那些遵守禁令的人。但是，我们发现某些静摩擦解在滑移发作的子区域中是向前非唯一的。与前向唯一的常规静摩擦解决方案相比，我们将这些解决方案称为单数。为了进一步了解非唯一动力学，我们随后介绍了该模型的一般正则化。这给出了一个奇摄动的问题，该问题捕获了原始不连续问题的主要特征。使用几何奇异摄动理论，我们确定了一种排斥缓慢的歧管，该歧管将向前的滑移与向前的粘滞解决方案分开，从而导致对初始条件的高度敏感性。在这个缓慢的流形上，我们发现了具有延迟滑移发生的物理解释的鸭式轨迹。最有趣的是 我们发现这些新解与分段平滑极限中的静力解不对应，因此看似非物理的，但它们很健壮，并且在我们考虑的正则化类别中普遍存在。最后，我们证明了正规化问题具有与卡纳德相互作用的一族周期性轨道。我们观察到，该族具有鞍形稳定性，并且在刚体极限内连接了不连续问题的两个规则的，粘滑的分支，这些分支否则会断开连接。