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

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.