标题： 非线性软化作为气候临界点的预测前兆 显示英文标题

标题： Nonlinear softening as a predictive precursor to climate tipping

摘要： 接近一种危险的分岔，从这种分岔开始，一个动力系统如地球的气候将跳转（倾斜）到不同的状态，当前的稳定状态位于一个逐渐缩小的吸引盆地内。在存在噪声扰动的情况下，该状态的持续性变得越来越危险。我们考虑一个潜在的势能，理论上定义为鞍点折叠，并通过平均法定义为霍普夫分岔。在接近稳定状态时，这个势能具有抛物线形式；但接近跳转时，它越来越受到软化非线性的影响。如果我们已经检测到线性衰减率的下降，非线性信息可以帮助我们估计由于噪声导致早期倾斜的可能性。我们认为需要从时间序列中提取关于潜在势能的非线性特征（“软化”）的信息，以判断倾斜的概率和时间。如果已经检测到线性衰减率的下降，这种分析是逻辑上的下一步。如果线性分析中没有明显的趋势，非线性软化在显示接近倾斜的接近程度方面甚至更加重要。经过广泛的正规形式校准研究后，我们检查了两个来自古气候倾斜事件的地质时间序列，以检测潜在井的软化情况。对于最后一次冰河时代结束的情况，我们发现没有令人信服的线性前兆，但我们识别出向温度升高显著的非线性软化。因此，该分析成功检测到了即将发生倾斜事件的预警。 显示英文摘要

摘要： Approaching a dangerous bifurcation, from which a dynamical system such as the Earth's climate will jump (tip) to a different state, the current stable state lies within a shrinking basin of attraction. Persistence of the state becomes increasingly precarious in the presence of noisy disturbances. We consider an underlying potential, as defined theoretically for a saddle-node fold and (via averaging) for a Hopf bifurcation. Close to a stable state, this potential has a parabolic form; but approaching a jump it becomes increasingly dominated by softening nonlinearities. If we have already detected a decrease in the linear decay rate, nonlinear information allows us to estimate the propensity for early tipping due to noise. We argue that one needs to extract information about the nonlinear features (a "softening") of the underlying potential from the time series to judge the probability and timing of tipping. This analysis is the logical next step if one has detected a decrease of the linear decay rate. If there is no discernable trend in the linear analysis, nonlinear softening is even more important in showing the proximity to tipping. After extensive normal form calibration studies, we check two geological time series from paleo-climate tipping events for softening of the underlying well. For the ending of the last ice age, where we find no convincing linear precursor, we identify a statistically significant nonlinear softening towards increasing temperature. The analysis has thus successfully detected a warning of the imminent tipping event.