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Rate-Induced Tipping and Saddle-Node Bifurcation for Quadratic Differential Equations with Nonautonomous Asymptotic Dynamics
SIAM Journal on Applied Dynamical Systems ( IF 2.1 ) Pub Date : 2021-03-18 , DOI: 10.1137/20m1339003
Iacopo P. Longo , Carmen Nún͂ez , Rafael Obaya , Martin Rasmussen

SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 500-540, January 2021.
An in-depth analysis of nonautonomous bifurcations of saddle-node type for scalar differential equations $x'=-x^2+q(t)\,x+p(t)$, where $q\colon\mathbb R\to\mathbb R$ and $p\colon\mathbb R\to\mathbb R$ are bounded and uniformly continuous, is fundamental to explaining the absence or occurrence of rate-induced tipping for the differential equation $y' =(y-(2/\pi)\arctan(ct))^2+p(t)$ as the rate $c$ varies on $[0,\infty)$. A classical attractor-repeller pair, whose existence for $c=0$ is assumed, may persist for any $c>0$, or disappear for a certain critical rate $c=c_0$, giving rise to rate-induced tipping. A suitable example demonstrates that one can have more than one critical rate, and the existence of the classical attractor-repeller pair may return when $c$ increases.


中文翻译:

具有非自治渐近动力学的二次微分方程速率诱导的倾斜和鞍节点分支

SIAM应用动力系统杂志,第20卷,第1期,第500-540页,2021年1月。
深入分析标量微分方程$ x'=-x ^ 2 + q(t)\,x + p(t)$的鞍节点类型的非自治分叉,其中$ q \ colon \ mathbb R \ to \ mathbb R $和$ p \ colon \ mathbb R \ to \ mathbb R $有界且一致连续,对于解释微分方程$ y'=(y-(2 / \ pi)\ arctan(ct))^ 2 + p(t)$,因为比率$ c $随$ [0,\ infty)$的变化而变化。假设存在一对经典吸引子-排斥子对,其对价为$ c = 0 $,可能会在任何$ c> 0 $时保持不变,或者对于某个临界速率$ c = c_0 $消失,从而引起费率诱发的小费。一个合适的例子表明,一个临界率可以不止一个,并且当$ c $增加时,经典的吸引-排斥对可能再次存在。
更新日期:2021-03-18
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