Critical transitions in piecewise uniformly continuous concave quadratic ordinary differential equations

TL;DR

This paper introduces a new analytical tool to detect critical transitions in concave quadratic scalar ODEs with piecewise uniformly continuous coefficients, highlighting how small parameter changes can cause dramatic shifts in system dynamics.

Contribution

The paper develops a novel method for analyzing critical transitions in nonautonomous ODEs with piecewise uniformly continuous coefficients, extending understanding of bifurcation phenomena.

Findings

The tool successfully identifies parameter values leading to critical transitions.

Numerical experiments demonstrate the method's applicability.

The approach clarifies the dynamics near bifurcation points.

Abstract

A critical transition for a system modelled by a concave quadratic scalar ordinary differential equation occurs when a small variation of the coefficients changes dramatically the dynamics, from the existence of an attractor-repeller pair of hyperbolic solutions to the lack of bounded solutions. In this paper, a tool to analyze this phenomenon for asymptotically nonautonomous ODEs with bounded uniformly continuous or bounded piecewise uniformly continuous coefficients is described, and used to determine the occurrence of critical transitions for certain parametric equations. Some numerical experiments contribute to clarify the applicability of this tool.