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.