English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider a dynamical system undergoing a saddle-node bifurcation with anexplicitly time dependent parameter~$p(t)$. The combined dynamics can beconsidered as a dynamical systems where $p$ is a slowly evolving parameter.Here, we investigate settings where the parameter features an overshoot. Itcrosses the bifurcation threshold for some finite duration $t_e$ and up to anamplitude $R$, before it returns to its initial value. We denote the overshootas safe when the dynamical system returns to its initial state. Otherwise, oneencounters runaway trajectories (tipping), and the overshoot is unsafe. Forshallow overshoots (small $R$) safe and unsafe overshoots are discriminated byan inverse square-root border, $t_e \propto R^{-1/2}$, as reported in earlierliterature. However, for larger overshoots we here establish a crossover toanother 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 weprovide examples for overshoots with exponents in the range $[-1, -1/2]$. Allresults are substantiated by numerical simulations, and it is discussed how theanalytic and numeric results pave the way towards improved risks assessmentsseparating safe from unsafe overshoots in climate, ecology and nonlineardynamics.

Constraining safe and unsafe overshoots in saddle-node bifurcations