Department of Mathematics, University of Portsmouth, Lion Gate Building, Lion Terrace, Portsmouth PO1 3HF, UK.
Philos Trans A Math Phys Eng Sci. 2012 Mar 13;370(1962):1205-27. doi: 10.1098/rsta.2011.0372.
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 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 discernible 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 palaeo-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.
当一个动力学系统（如地球气候）即将到达一个危险的分岔点并跳跃（转折）到一个不同的状态时，当前的稳定状态处于一个吸引力不断缩小的盆地中。在存在噪声干扰的情况下，状态的持续存在变得越来越不稳定。我们认为，人们需要从时间序列中提取关于潜在势的非线性特征（“软化”）的信息，以判断转折的概率和时间。如果已经检测到线性衰减率的降低，那么这种分析就是合乎逻辑的下一步。如果在线性分析中没有明显的趋势，那么非线性软化对于显示接近转折的程度就更加重要。在进行了广泛的规范形式校准研究之后，我们检查了古气候转折事件的两个地质时间序列，以检查潜在基准的软化情况。对于上一个冰河时代的结束，我们没有发现令人信服的线性前兆，而是确定了一个朝着温度升高方向的具有统计学意义的非线性软化。
