Centre for Systems, Dynamics and Control, College of Engineering, Mathematics and Physical Sciences, Harrison Building, University of Exeter, Exeter EX4 4QF, United Kingdom.
EPSRC Centre for Predictive Modelling in Healthcare, University of Exeter, Exeter EX4 4QJ, United Kingdom.
Phys Rev E. 2017 May;95(5-1):052209. doi: 10.1103/PhysRevE.95.052209. Epub 2017 May 12.
We propose an approximation for the probability of tipping when the speed of parameter change and additive white noise interact to cause tipping. Our approximation is valid for small to moderate drift speeds and helps to estimate the probability of false positives and false negatives in early-warning indicators in the case of rate- and noise-induced tipping. We illustrate our approximation on a prototypical model for rate-induced tipping with additive noise using Monte Carlo simulations. The formula can be extended to close encounters of rate-induced tipping and is otherwise applicable to other forms of tipping. We also provide an asymptotic formula for the critical ramp speed of the parameter in the absence of noise for a general class of systems undergoing rate-induced tipping.
我们提出了一种当参数变化速度和加性白噪声相互作用导致 tipping 时 tipping 概率的近似值。我们的近似值对于小到中等的漂移速度有效，有助于在由于速率和噪声引起的 tipping 的情况下估计早期预警指标中的假阳性和假阴性的概率。我们使用蒙特卡罗模拟在带有加性噪声的典型速率诱导 tipping 模型上对我们的近似值进行了说明。该公式可以扩展到速率诱导 tipping 的近距离遭遇，否则适用于其他形式的 tipping。我们还为一般类别的经历速率诱导 tipping 的系统提供了在没有噪声的情况下参数的临界斜坡速度的渐近公式。
