There is a growing awareness that catastrophic phenomena in biology and medicine can be mathematically represented in terms of saddle–node bifurcations. In particular, the term “tipping”, or critical transition has in recent years entered the discourse of the general public in relation to ecology, medicine, and public health. The saddle–node bifurcation and its associated theory of catastrophe as put forth by Thom and Zeeman has seen applications in a wide range of fields including molecular biophysics, mesoscopic physics, and climate science. In this paper, we investigate a simple model of a non-autonomous system with a time-dependent parameter $p\left(\tau \right)$ and its corresponding “dynamic” (time-dependent) saddle–node bifurcation by the modern theory of non-autonomous dynamical systems. We show that the actual point of no return for a system undergoing tipping can be significantly delayed in comparison to the breaking time $\hat{\tau }$ at which the corresponding autonomous system with a time-independent parameter $p_a=p\left(\hat{\tau }\right)$ undergoes a bifurcation. A dimensionless parameter $\alpha =\lambda p_0^3{V}^{-2}$ is introduced, in which $\lambda$ is the curvature of the autonomous saddle–node bifurcation according to parameter $p\left(\tau \right)$, which has an initial value of $p_0$ and a constant rate of change $V$. We find that the breaking time $\hat{\tau }$ is always less than the actual point of no return ${\tau }^{\ast }$ after which the critical transition is irreversible; specifically, the relation ${\tau }^{\ast }-\hat{\tau }\simeq 2.338{\left(\lambda V\right)}^{-\frac{1}{3}}$ is analytically obtained. For a system with a small $\lambda V$, there exists a significant window of opportunity $(\hat{\tau },{\tau }^{\ast })$ during which rapid reversal of the environment can save the system from catastrophe.

人们越来越认识到，生物学和医学上的灾难性现象可以用鞍结分叉来数学表示。特别是，近年来，“小费”或“关键过渡”一词已进入公众关于生态，医学和公共卫生的讨论。Thom和Zeeman提出的鞍节点分叉及其相关的灾难理论已经在分子生物学，介观物理学和气候科学等广泛领域得到应用。在本文中，我们研究具有时间相关参数的非自治系统的简单模型$p（\tau ）$非现代动力系统的现代理论及其相应的“动力”（随时间变化的）鞍节点分叉。我们显示，与中断时间相比，发生倾翻的系统的实际不返回点会大大延迟 $\hat{\tau }$ 相应的具有时间独立参数的自治系统 $p_{\mathrm{一种}}=p（\hat{\tau }）$经历分叉。无量纲参数$\alpha =\lambda p_0^3{V}^{-2}$ 介绍，其中 $\lambda$ 是根据参数的自主鞍节点分叉的曲率 $p（\tau ）$，其初始值为 $p_0$ 和恒定的变化率 $V$。我们发现休息时间$\hat{\tau }$ 总是小于实际的无收益点 ${\tau }^{\ast }$此后的关键转变是不可逆的；具体来说，关系${\tau }^{\ast }-\hat{\tau }\simeq 2。338{（\lambda V）}^{-\frac{\mathrm{1个}}{3}}$通过分析获得。对于小系统$\lambda V$，存在很大的机会之窗 $（\hat{\tau }，{\tau }^{\ast }）$ 在此期间，环境的快速逆转可以使系统免于灾难。