Generalized Pitchfork Bifurcations in D-Concave Nonautonomous Scalar Ordinary Differential Equations

TL;DR

This paper investigates global bifurcation diagrams in nonautonomous scalar ODEs with specific perturbations, revealing a new type of pitchfork bifurcation unique to nonautonomous systems, characterized by changes in minimal sets and attractor shapes.

Contribution

It introduces and extensively studies a novel global generalized pitchfork bifurcation in nonautonomous scalar ODEs, absent in autonomous dynamics.

Findings

Identification of bifurcations via minimal sets and attractor shape changes

Discovery of a new global generalized pitchfork bifurcation pattern

Analysis of bifurcation behavior under different perturbations

Abstract

The global bifurcation diagrams for two different one-parametric perturbations ($+\lambda x$ and $+\lambda x^2$) of a dissipative scalar nonautonomous ordinary differential equation $x'=f(t,x)$ are described assuming that 0 is a constant solution, that $f$ is recurrent in $t$, and that its first derivative with respect to $x$ is a strictly concave function. The use of the skewproduct formalism allows us to identify bifurcations with changes in the number of minimal sets and in the shape of the global attractor. In the case of perturbation $+\lambda x$, a so-called global generalized pitchfork bifurcation may arise, with the particularity of lack of an analogue in autonomous dynamics. This new bifurcation pattern is extensively investigated in this work.