Saddle-node bifurcations for concave in measure and d-concave in measure skewproduct flows with applications to population dynamics and circuits

TL;DR

This paper explores saddle-node bifurcations in nonautonomous scalar ODEs with concave in measure properties, providing a framework for analyzing complex dynamical behaviors and critical transitions in models including population dynamics and circuits.

Contribution

It introduces a generalized bifurcation framework for concave in measure and d-concave in measure equations, extending analysis to highly chaotic and stochastic-like systems.

Findings

Describes simple and double saddle-node bifurcation diagrams for these equations.

Framework applies to almost stochastic equations with chaotic coefficients.

Analyzes critical transitions in diverse models beyond previous methods.

Abstract

Concave in measure and d-concave in measure nonautonomous scalar ordinary differential equations given by coercive and time-compactible maps have similar properties to equations satisfying considerably more restrictive hypotheses. This paper describes the generalized simple or double saddle-node bifurcation diagrams for one-parametric families of equations of these types, from which the dynamical possibilities for each of the equations follow. This new framework allows the analysis of ``almost stochastic" equations, whose coefficients vary in very large chaotic sets. The results also apply to the analysis of the occurrence of critical transitions for a range of models much larger than in previous approaches.