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

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.