约束鞍结分岔中的安全与不安全过冲
Constraining safe and unsafe overshoots in saddle-node bifurcations
AI总结：
本文研究显式时变参数下鞍结分岔的过冲安全性，揭示浅过冲到深过冲时安全边界幂律由 $R^{-1/2}$ 向依赖参数渐近形式的指数交叉，并以模拟验证。
摘要：
我们考虑一个正在经历鞍结分岔的动力系统，其参数 $p(t)$ 显式依赖于时间。这一组合动力学可被视为一个 $p$ 为缓慢演化参数的动力系统。本文研究参数具有过冲特征的情形：参数在有限持续时间 $t_e$ 内越过分岔阈值，过冲幅度达到 $R$，随后回到其初始值。当动力系统回到初始状态时，我们称该过冲是安全的；否则会出现逃逸轨迹（临界转变），该过冲即为不安全。对于浅过冲（小 $R$），安全与不安全过冲由平方根倒数边界 $t_e \propto R^{-1/2}$ 区分，这与早期文献一致。然而，对于更大的过冲，我们在此确立了向另一种幂律的交叉，其指数取决于 $p(t)$ 的渐近行为。对于具有有限支撑的过冲，我们发现 $t_e \propto R^{-1}$，并给出了指数位于 $[-1,-1/2]$ 范围内的过冲示例。所有结果均由数值模拟证实；文中还讨论了解析与数值结果如何为改进风险评估铺平道路，从而区分气候、生态和非线性动力学中的安全与不安全过冲。
英文摘要：
We consider a dynamical system undergoing a saddle-node bifurcation with an explicitly time dependent parameter~$p(t)$. The combined dynamics can be considered as a dynamical systems where $p$ is a slowly evolving parameter. Here, we investigate settings where the parameter features an overshoot. It crosses the bifurcation threshold for some finite duration $t_e$ and up to an amplitude $R$, before it returns to its initial value. We denote the overshoot as safe when the dynamical system returns to its initial state. Otherwise, one encounters runaway trajectories (tipping), and the overshoot is unsafe. For shallow overshoots (small $R$) safe and unsafe overshoots are discriminated by an inverse square-root border, $t_e \propto R^{-1/2}$, as reported in earlier literature. However, for larger overshoots we here establish a crossover to another power law with an exponent that depends on the asymptotics of $p(t)$. For overshoots with a finite support we find that $t_e \propto R^{-1}$, and we provide examples for overshoots with exponents in the range $[-1, -1/2]$. All results are substantiated by numerical simulations, and it is discussed how the analytic and numeric results pave the way towards improved risks assessments separating safe from unsafe overshoots in climate, ecology and nonlinear dynamics.
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