Critical transitions for scalar nonautonomous systems with concave nonlinearities: some rigorous estimates

TL;DR

This paper provides rigorous criteria for predicting critical transitions in nonautonomous scalar differential equations with concave nonlinearities, focusing on tracking and tipping phenomena, supported by computer simulations.

Contribution

It extends the understanding of global dynamics in nonautonomous scalar ODEs with concave nonlinearities and offers criteria to identify critical transitions without numerical attractor approximation.

Findings

Criteria for tracking and tipping are rigorously established.

Simulations confirm the accuracy of the theoretical estimates.

The approach avoids reliance on numerical approximation of attractors.

Abstract

The global dynamics of a nonautonomous Carath\'eodory scalar ordinary differential equation $x'=f(t,x)$, given by a function $f$ which is concave in $x$, is determined by the existence or absence of an attractor-repeller pair of hyperbolic solutions. This property, here extended to a very general setting, is the key point to classify the dynamics of an equation which is a transition between two nonautonomous asypmtotic limiting equations, both with an attractor-repeller pair. The main focus of the paper is to get rigorous criteria guaranteeing tracking (i.e., connection between the attractors of the past and the future) or tipping (absence of connection) for the particular case of equations $x'=f(t,x-\Gamma(t))$, where $\Gamma$ is asymptotically constant. Some computer simulations show the accuracy of the obtained estimates, which provide a powerful way to determine the occurrence of critical transitions without relying on a numerical approximation of the (always existing) locally pullback attractor.