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An in-depth analysis of nonautonomous bifurcations of saddle-node type forscalar differential equations $x'=-x^2+q(t)\,x+p(t)$, where$q\colon\mathbb{R}\to\mathbb{R}$ and $p\colon\mathbb{R}\to\mathbb{R}$ arebounded and uniformly continuous, is fundamental to explain the absence oroccurrence of rate-induced tipping for the differential equation $y'=(y-(2/\pi)\arctan(ct))^2+p(t)$ as the rate $c$ varies on $[0,\infty)$. Aclassical attractor-repeller pair, whose existence for $c=0$ is assumed, maypersist for any $c>0$, or disappear for a certain critical rate $c=c_0$, givingrise to rate-induced tipping. A suitable example demonstrates that one can havemore than one critical rate, and the existence of the classicalattractor-repeller pair may return when $c$ increases.

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