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： 接近一个危险的分岔，从该分岔点开始，如地球气候这样的动态系统将跳（倾斜）到一个不同的状态，当前稳定状态位于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不稳定。 我们考虑一个潜在的理论定义，用于鞍点折叠，并通过平均方法用于霍普夫分岔。 靠近稳定状态时，这个势能具有抛物线形式；但接近跳跃时，它越来越由软化非线性主导。 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倾向。 如果线性分析中没有明显趋势，非线性软化在显示接近倾斜的接近程度方面就更加重要。 经过广泛的正规型校准研究后，我们将该技术应用于两个来自古气候倾斜事件的地质时间序列。 对于上一个冰河时代结束的情况，我们发现没有令人信服的线性前兆，但我们识别出统计显著的非线性软化，朝着温度上升的方向。 因此，该分析成功地检测到了即将发生的倾斜事件的预警。 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. 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 apply our technique to two geological time series from paleo-climate tipping events. For the ending of the last ice age, where we find is 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.