Ritchie Paul, Karabacak Özkan, Sieber Jan
Earth System Science, College of Life and Environmental Sciences, Harrison Building, University of Exeter, Exeter EX4 4QF, UK.
Department of Electronic Systems, Automation and Control, Aalborg University, Fredrik Bajers Vej 7 C, Aalborg East 9220, Denmark.
Proc Math Phys Eng Sci. 2019 Feb;475(2222):20180504. doi: 10.1098/rspa.2018.0504. Epub 2019 Feb 27.
A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold tipping point, caused by a run-away positive feedback loop. We study what happens if one turns around after one has crossed the threshold. We derive a simple criterion that relates how far the parameter exceeds the tipping threshold maximally and how long the parameter stays above the threshold to avoid tipping in an inverse-square law to observable properties of the dynamical system near the fold. For the case when the dynamical system is subject to stochastic forcing we give an approximation to the probability of tipping if a parameter changing in time reverses near the tipping point. The derived approximations are valid if the parameter change in time is sufficiently slow. We demonstrate for a higher-dimensional system, a model for the Indian summer monsoon, how numerically observed escape from the equilibrium converge to our asymptotic expressions. The inverse-square law between peak of the parameter forcing and the time the parameter spends above a given threshold is also visible in the level curves of equal probability when the system is subject to random disturbances.
一个典型的翻转情形是，一个动力系统在一个逃逸正反馈回路的作用下，经历一个缓慢的参数漂移并越过一个折叠翻转点。我们研究如果在越过阈值后折返会发生什么。我们推导了一个简单的准则，该准则将参数超过翻转阈值的最大程度以及参数保持在阈值之上的时间与动力系统在折叠附近的可观测性质以反平方定律联系起来，以避免翻转。对于动力系统受到随机强迫的情况，如果一个随时间变化的参数在翻转点附近反转，我们给出了翻转概率的近似值。如果参数随时间的变化足够缓慢，所推导的近似值就是有效的。我们针对一个高维系统——印度夏季风模型，展示了数值观测到的从平衡态的逃逸如何收敛到我们的渐近表达式。当系统受到随机扰动时，在等概率的等高线中也能看到参数强迫峰值与参数在给定阈值之上花费的时间之间的反平方定律。
