Warning signs for tipping points (or critical transitions) have been very actively studied. Although the theory has been applied successfully in models and in experiments, for many complex systems, e.g., for tipping in climate systems, there are ongoing debates, when warning signs can be extracted from data. In this work, we provide an explanation, why these difficulties occur, and we significantly advance the general theory of warning signs for nonlinear stochastic dynamics. A key scenario deals with stochastic systems approaching a bifurcation point dynamically upon slow parameter variation. The stochastic fluctuations are generically able to probe the dynamics near a deterministic attractor to detect critical slowing down. Using scaling laws, one can then anticipate the distance to a bifurcation. Previous warning signs results assume that the noise is Markovian, most often even white. Here we study warning signs for non-Markovian systems including colored noise and $\alpha$-regular Volterra processes (of which fractional Brownian motion and the Rosenblatt process are special cases). We prove that early-warning scaling laws can disappear completely or drastically change their exponent based upon the parameters controlling the noise process. This provides a clear explanation, why applying standard warning signs results to reduced models of complex systems may not agree with data-driven studies. We demonstrate our results numerically in the context of a box model of the Atlantic Meridional Overturning Circulation (AMOC).
翻译:对临界点(或关键过渡)的警告信号进行了非常积极的研究。尽管理论在模型和实验中已经成功地应用,但对于许多复杂的系统,例如气候系统,理论已经成功地应用到模型和实验中,对于许多复杂的系统,例如对于气候系统来说,理论已经应用得非常成功,但在从数据中可以提取警告信号时,正在进行辩论。在这项工作中,我们提供了一个解释,为什么这些困难会发生,而且我们大大推进非线性随机动态的警告信号一般理论。一个关键的设想是,在参数变化缓慢时,随机系统系统会以动态的方式接近两极点。一般而言,随机波动能够探测到确定性吸引者附近的动态,以探测临界减慢速度的系统。我们用比例法来预测离两极点的距离。之前的警告信号结果假定噪音是马尔科文,最经常是白色的。我们在这里研究非马尔科文系统警告信号,包括彩色噪音和美元固定的伏特拉进程(其中分数的布朗运动和罗森布拉特进程是特殊的例子)。我们证明,预警缩缩缩缩缩图法的模型可以彻底地消失或急剧改变其方向标志。我们使用标准的轨道模型的模型,从而可以解释其精确的系统。我们的标准反向后的结果。我们用一个精确地研究。我们用的方法来解释。我们用的方法可以解释。