具有速率和状态摩擦的模型中的一类新的弛豫振荡
A new type of relaxation oscillation in a model with rate-and-state friction
摘要：
本文证明了具有Ruina速率-状态摩擦律的单块Burridge-Knopoff模型中一类新的弛豫振荡的存在性。在相关参数区域，系统是两个慢变量和一个快变量的慢-快系统。这种振荡的特殊性有以下几个原因：第一，它的奇异极限是无界的，周期的振幅像$\log ε^{-1}$一样增长为$ε\rightarrow 0$。作为这种估计的结果，循环的无界性不能通过变量的简单的$ε$依赖缩放来捕获，参见例如\cite{Gucwa2009783}。因此，我们得到它在庞加莱球面上的极限。在这里，我们发现，奇异极限由一个吸引的临界流形上的慢的一部分，和一个快速的一部分上的赤道（即在$\infty$）的庞加莱球，其中包括沿中心流形的运动。在这个中心流形上的减少的流动以一种特殊的方式沿着流形的边界流出，导致复杂的返回到慢流形。我们证明了存在的极限环的返回映射是一个压缩。这一部分的主要技术难点是由于临界流形在无穷远处以指数速率失去双曲性。因此，我们在\cite{kristiansen2017a}中使用该方法，在扩展的相空间中应用标准爆破技术。在这种方式中，我们确定一个奇异的周期，由$12$件，所有与理想的双曲性质，使扰动到一个实际的极限环$0<ε\ll 1$。
英文摘要：
In this paper we prove the existence of a new type of relaxation oscillation occurring in a one-block Burridge-Knopoff model with Ruina rate-and-state friction law. In the relevant parameter regime, the system is slow-fast with two slow variables and one fast. The oscillation is special for several reasons: Firstly, its singular limit is unbounded, the amplitude of the cycle growing like $\log ε^{-1}$ as $ε\rightarrow 0$. As a consequence of this estimate, the unboundedness of the cycle cannot be captured by a simple $ε$-dependent scaling of the variables, see e.g. \cite{Gucwa2009783}. We therefore obtain its limit on the Poincaré sphere. Here we find that the singular limit consists of a slow part on an attracting critical manifold, and a fast part on the equator (i.e. at $\infty$) of the Poincaré sphere, which includes motion along a center manifold. The reduced flow on this center manifold runs out along the manifold's boundary, in a special way, leading to a complex return to the slow manifold. We prove the existence of the limit cycle by showing that a return map is a contraction. The main technical difficulty in this part is due to the fact that the critical manifold loses hyperbolicity at an exponential rate at infinity. We therefore use the method in \cite{kristiansen2017a}, applying the standard blowup technique in an extended phase space. In this way we identify a singular cycle, consisting of $12$ pieces, all with desirable hyperbolicity properties, that enables the perturbation into an actual limit cycle for $0<ε\ll 1$.
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