Imperfect bifurcations via topological methods in superlinear indefinite problems

TL;DR

This paper investigates how the bifurcation structures of superlinear indefinite problems change when symmetry is broken, revealing the emergence and breakdown of complex solution branches through topological methods.

Contribution

It provides a detailed analysis of the transition from symmetric to asymmetric bifurcation diagrams using Poincaré maps and topological techniques.

Findings

Bifurcation diagrams become disconnected with asymmetric weights.

Secondary bifurcations break as symmetry is perturbed.

Construction of bifurcation diagrams via Poincaré maps.

Abstract

In [5] the structure of the bifurcation diagrams of a class of superlinear indefinite problems with a symmetric weight was ascertained, showing that they consist of a primary branch and secondary loops bifurcating from it. In [4] it has been proved that, when the weight is asymmetric, the bifurcation diagrams are no longer connected since parts of the primary branch and loops of the symmetric case form an arbitrarily high number of isolas. In this work we give a deeper insight on this phenomenon, studying how the secondary bifurcations break as the weight is perturbed from the symmetric situation. Our proofs rely on the approach of [5,4], i.e. on the construction of certain Poincar\'e maps and the study of how they vary as some of the parameters of the problems change, constructing in this way the bifurcation diagrams.