Rate-induced tipping and saddle-node bifurcation for quadratic differential equations with nonautonomous asymptotic dynamics
Iacopo P. Longo, Carmen Núñez, Rafael Obaya, Martin Rasmussen
Abstract
An in-depth analysis of nonautonomous bifurcations of saddle-node\n\t\t\t\t type for scalar differential equations $x'=-x^2+q(t)\\,x+p(t)$,\n\t\t\t\t where $q\\colon\\R\\to\\R$ and $p\\colon\\R\\to\\R$ are bounded and uniformly\n\t\t\t\t continuous, is fundamental to explain the absence or occurrence of\n\t\t\t\t rate-induced tipping for the differential equation\n\t\t\t\t $y' =(y-(2/\\pi)\\arctan(ct))^2+p(t)$ as the rate $c$ varies on $[0,\\infty)$.\n\t\t\t\t A classical attractor-repeller pair, whose existence for $c=0$ is assumed,\n\t\t\t\t may persist for any $c>0$, or disappear for a certain critical rate $c=c_0$,\n\t\t\t\t giving rise to rate-induced tipping. A suitable example demonstrates that\n\t\t\t\t this tipping phenomenon may be reversible.