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

Abstract

An in-depth analysis of nonautonomous bifurcations of saddle-node type for scalar differential equations $x^{\prime }=-x^2+q(t)x+p(t)$, where $q:\mathbb{R}\to \mathbb{R}$ and $p:\mathbb{R}\to \mathbb{R}$ are bounded and uniformly continuous, is fundamental to explain the absence or occurrence of rate-induced tipping for the differential equation $y^{\prime }=(y-(2/\pi )\arctan (ct){)}^2+p(t)$ 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.

40 pages, 10 figures

References in corpus (1)

• Basin bifurcations, oscillatory instability and rate-induced thresholds for AMOC in a global oceanic box model