具有首阶非线性项的临界点分析及其可预报性
Analysis and Predictability for Tipping Points with Leading-Order Nonlinear Terms
摘要：
临界点在各种应用中以及从数学的角度都得到了积极的研究。从理论上理解临界点预警信号的一个主要技术是使用快慢随机微分方程的框架。在许多关于临界点之前存在方差和自相关增长的论点中，一个关键假设是使用线性化论点，即，控制随机微分方程的确定性（或漂移）部分的首阶项是线性的。这个假设保证了一个局部近似通过Ornstein-Uhlenbeck过程在正常双曲制度之前，但充分有界远离，分歧。在本文中，我们将这种情况推广到首阶非线性项的一个快速变量的设置。我们工作在准稳定的制度，并证明了快速的变量有一个明确的平稳分布，我们计算的方差作为一个分叉引起的临界点接近的标度律。我们交叉验证的标度律数值。此外，我们提供了一个计算研究的可预测性使用预警信号的领导阶非线性项的基础上接收操作员的特征曲线。
英文摘要：
Tipping points have been actively studied in various applications as well as from a mathematical viewpoint. A main technique to theoretically understand early-warning signs for tipping points is to use the framework of fast-slow stochastic differential equations. A key assumption in many arguments for the existence of variance and auto-correlation growth before a tipping point is to use a linearization argument, i.e., the leading-order term governing the deterministic (or drift) part of stochastic differential equation is linear. This assumption guarantees a local approximation via an Ornstein-Uhlenbeck process in the normally hyperbolic regime before, but sufficiently bounded away from, a bifurcation. In this paper, we generalize the situation to leading-order nonlinear terms for the setting of one fast variable. We work in the quasi-steady regime and prove that the fast variable has a well-defined stationary distribution and we calculate the scaling law for the variance as a bifurcation-induced tipping point is approached. We cross-validate the scaling law numerically. Furthermore, we provide a computational study for the predictability using early-warning signs for leading-order nonlinear terms based upon receiver-operator characteristic curves.
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