Bifurcation Theory of Attractors and Minimal Sets in D-Concave Nonautonomous Scalar Ordinary Differential Equations

TL;DR

This paper investigates bifurcation phenomena in scalar nonautonomous ODEs with concave derivatives, analyzing how minimal sets and attractors change abruptly at bifurcation points using skewproduct formalism.

Contribution

It introduces a novel analysis of bifurcations in nonautonomous scalar ODEs considering concavity and coercivity, highlighting the shape and properties of attractors and minimal sets.

Findings

Identification of saddle-node, transcritical, and pitchfork bifurcations in minimal sets.

Characterization of global attractor discontinuities.

Analysis of minimal set properties under parameter variation.

Abstract

Two one-parametric bifurcation problems for scalar nonautonomous ordinary differential equations are analyzed assuming the coercivity of the time-dependent function determining the equation and the concavity of its derivative with respect to the state variable. The skewproduct formalism leads to the analysis of the number and properties of the minimal sets and of the shape of the global attractor, whose abrupt variations determine the occurrence of local saddle-node, local transcritical and global pitchfork bifurcation points of minimal sets and of discontinuity points of the global attractor.