Path-Connectedness in Global Bifurcation Theory

TL;DR

This paper presents examples demonstrating that global bifurcation sets can be disconnected or consist of singleton components, challenging common assumptions in bifurcation theory.

Contribution

It constructs explicit examples showing that global bifurcation sets may be disconnected or trivial, even under classical hypotheses.

Findings

Global bifurcation sets can be disconnected.

Connected components of bifurcation sets may be singletons.

Bifurcation sets may lack path-connectedness despite classical conditions.

Abstract

A celebrated result in bifurcation theory is that global connected sets of non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues of odd algebraic multiplicity of the linearized problem when the operators involved are compact. In this paper a simple example is constructed which satisfies the regularity hypotheses of the global bifurcation theorem, and the eigenvalue has algebraic multiplicity one, yet all the path-connected components of the connected sets that bifurcate are singletons. Another example shows that even when the operators are everywhere infinitely differentiable and classical bifurcation occurs locally at a simple eigenvalue, the global continuum may not be path-connected away from the bifurcation point. A third example shows that the non-trivial solutions which, by variational theory, bifurcate from eigenvalues of any multiplicity when the problem has gradient structure, may not be connected and may contain no paths except singletons.