SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2371-2402, January 2020.
We discuss tipping phenomena in nonautonomous systems using an example of a bistable ecosystem model with environmental changes represented by time-varying parameters [Scheffer et al., Ecosystems, 11 (2008), pp. 275--279]. We give simple testable criteria for the occurrence of nonautonomous tipping from the herbivore-dominating equilibrium to the plant-only equilibrium using global properties of the autonomous frozen system with fixed-in-time parameters. To begin with, we use classical autonomous bifurcation analysis to identify a codimension-three degenerate Bogdanov--Takens bifurcation: the source of a dangerous subcritical Hopf bifurcation and the organizing center for bifurcation-induced tipping (B-tipping). Then, we introduce the concept of basin instability for equilibria to identify parameter paths along which genuine nonautonomous rate-induced tipping (R-tipping) occurs without crossing any classical autonomous bifurcations. We explain nonautonomous R-tipping in terms of maximal canard trajectories and produce nonautonomous tipping diagrams in the plane of the magnitude and rate of a parameter shift to uncover intriguing R-tipping tongues and wiggling tipping-tracking bifurcation curves. Discussion of nontrivial dynamics arising from the interaction between B-tipping and R-tipping identifies “points of no return” where tipping cannot be prevented by the parameter trend reversal and “points of return tipping” where tipping is inadvertently induced by the parameter trend reversal. Our results give new insight into the sensitivity of ecosystems to the magnitudes and rates of environmental change. Finally, a comparison between “tilted” saddle-node and subcritical Hopf normal forms reveals some universal tipping properties due to basin instability, a generic dangerous bifurcation, or the combination of both.

中文翻译：

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小费现象和生态系统的无收益点：超越经典的分歧

SIAM应用动力系统杂志，第19卷，第4期，第2371-2402页，2020年1月。
我们使用一个双稳态生态系统模型的示例讨论非自治系统中的倾翻现象，其环境变化由时变参数表示[Scheffer et al。，Ecosystems，11（2008），pp.275--279]。我们使用具有固定时间参数的自动冻结系统的全局属性，给出了从草食动物占主导的平衡到仅植物平衡的非自主倾翻发生的简单可测试标准。首先，我们使用经典的自主分叉分析来确定共维数三个简并的Bogdanov-Takens分叉：危险的次临界Hopf分叉的来源以及分叉诱发的倾翻的组织中心。然后，我们引入了平衡盆地不稳定性的概念，以识别真正的非自治速率诱发的倾翻（R-tipping）沿其发生的参数路径，而不会跨越任何经典的自主分叉。我们以最大的卡纳德轨迹来解释非自治的R倾斜，并在参数移动的幅度和速率平面上生成非自治的倾斜图，以揭示引人入胜的R倾斜舌和摆动的倾斜跟踪分叉曲线。由B小费和R小费之间的相互作用引起的非平凡动力学的讨论确定了“无返回点”，其中参数趋势反转无法防止小费；“小费返回点”，其中参数趋势反转无意中导致了小费。 。我们的结果为生态系统对环境变化的幅度和速率的敏感性提供了新的见解。最后，将“倾斜”的鞍形节点和次临界霍普夫法线形式进行比较，发现由于盆地不稳，一般的危险分叉或两者兼而有之，具有一些普遍的倾翻特性。