Critical Transitions in Piecewise Uniformly Continuous Concave Quadratic Ordinary Differential Equations

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.

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

“…In this paper, we show that, if one of these equations is concave, then the only possible bifurcation is the nonautonomous saddle-node bifurcation of a pair of hyperbolic solutions, and that all the tipping points occurring for these equations are in fact bifurcations of such type. This extends the previous conclusions obtained in [32,33] for quadratic concave differential equations to this more general framework.…”

“…This property ensures that also the transition equation T ′ = g d t (t, T) has a strict lower solution in the area of concavity, and hence an atractor-repeller pair. In addition, using similar techniques to those of [32,33], it can be proved that this attractor-repeller pair approaches that of the past equation as time decreases and that of the future equation as time increases, which is the situation usually called tracking. And this is true for all the equations T ′ = g d c(t+l) (t, T) for c > 0 and l ∈ R. In other words: there are no critical transitions of rate-induced, size-induced or phase-induced type for d ∈ (5/9, d 2 ).…”

“…According to theorem 3.4, Case A is equivalent to the existence of an attractor-repeller pair. As in the quadratic case analysed in [32], much more can be said in any of the three situations if the next condition (assumed when indicated) holds for the 'unperturbed' equation (3.1) (i.e. for x ′ = f(t, x)): This section is devoted to describe these 'extra' properties, which relate the dynamics of (4.1) to those of its 'past' and 'future' instances,…”

“…Namely, when a time-dependent variation of parameters intervenes [23]. This is the case of rate-induced tipping [6,7,24,32,33,45], sizeinduced tipping [16,32], and phase-induced tipping [2,3,16,32].…”

“…Recently, an interpretation of these phenomena through nonautonomous bifurcation theory has arisen [15,17,32,33]. In this new formulation, all the terms involved (the past and the future systems and the transition equations between them) are nonautonomous.…”