Rate-Induced Tipping and Saddle-Node Bifurcation for Quadratic Differential Equations with Nonautonomous Asymptotic Dynamics
Iacopo P. Longo, Carmen Nún͂ez, Rafael Obaya, Martin Rasmussen
Abstract
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.