Rate-induced tipping and saddle-node bifurcation for quadratic differential equations with nonautonomous asymptotic dynamics

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 $q\colon\R\to\R$ and $p\colon\R\to\R$ are bounded and uniformly continuous, is fundamental to explain 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 this tipping phenomenon may be reversible.