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

TL;DR

This paper investigates how nonautonomous saddle-node bifurcations influence rate-induced tipping in quadratic differential equations with time-dependent parameters, revealing complex behaviors including multiple critical rates and attractor-repeller pair persistence.

Contribution

It provides a detailed analysis of nonautonomous bifurcations in quadratic equations, highlighting conditions for rate-induced tipping and demonstrating phenomena like multiple critical rates and attractor-repeller pair recurrence.

Findings

Existence of multiple critical rates for tipping.

Persistence of attractor-repeller pairs across rates.

Complex bifurcation behaviors in nonautonomous systems.

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 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 one can have more than one critical rate, and the existence of the classical attractor-repeller pair may return when $c$ increases.