Path-connectedness in global bifurcation theory

J. F. Toland; J. F. Toland

doi:10.3934/era.2021079

Electronic Research Archive

2021, Volume 29, Issue 6: 4199-4213. doi: 10.3934/era.2021079

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Path-connectedness in global bifurcation theory

J. F. Toland

Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

Received: 01 April 2021 Revised: 01 August 2021 Published: 08 October 2021
Primary: 47J15, 58E07, 35B32; Secondary: 54F15

A celebrated result in bifurcation theory is that, when the operators involved are compact, global connected sets of non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues of odd algebraic multiplicity of the linearized problem. This paper presents a simple example in which the hypotheses of the global bifurcation theorem are satisfied, 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 continua may not be path-connected away from the bifurcation point. A third example shows that the non-trivial solutions which bifurcate at non-zero eigenvalues, irrespective of multiplicity when the problem has gradient structure, may not be connected and may not contain any paths except singletons.
- Global bifurcation,
- point-set topology,
- path-connectedness,
- hereditarily indecomposable continua,
- topological degree,
- variational methods
Citation: J. F. Toland. Path-connectedness in global bifurcation theory[J]. Electronic Research Archive, 2021, 29(6): 4199-4213. doi: 10.3934/era.2021079

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Abstract

A celebrated result in bifurcation theory is that, when the operators involved are compact, global connected sets of non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues of odd algebraic multiplicity of the linearized problem. This paper presents a simple example in which the hypotheses of the global bifurcation theorem are satisfied, 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 continua may not be path-connected away from the bifurcation point. A third example shows that the non-trivial solutions which bifurcate at non-zero eigenvalues, irrespective of multiplicity when the problem has gradient structure, may not be connected and may not contain any paths except singletons.

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