A degenerate bifurcation from simple eigenvalue theorem

Ping Liu; Junping Shi; Ping Liu; Junping Shi

doi:10.3934/era.2022006

Electronic Research Archive

2022, Volume 30, Issue 1: 116-125. doi: 10.3934/era.2022006

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A degenerate bifurcation from simple eigenvalue theorem

Ping Liu ¹,
Junping Shi ^{2
,
,}

1.
Y. Y. Tseng Functional Analysis Research Center and School of Mathematical Sciences, Harbin Normal University, Harbin, Heilongjiang 150025, China
2.
Department of Mathematics, William & Mary, Williamsburg, VA 23187-8795, USA

Received: 26 August 2021 Revised: 16 November 2021 Accepted: 16 November 2021 Published: 13 December 2021

A new bifurcation from simple eigenvalue theorem is proved for general nonlinear functional equations. It is shown that in this bifurcation scenario, the bifurcating solutions are on a curve which is tangent to the line of trivial solutions, while in typical bifurcations the curve of bifurcating solutions is transversal to the line of trivial ones. The stability of bifurcating solutions can be determined, and examples from partial differential equations are shown to demonstrate such bifurcations.
- degenerate bifurcation,
- simple eigenvalue,
- tangential bifurcation
Citation: Ping Liu, Junping Shi. A degenerate bifurcation from simple eigenvalue theorem[J]. Electronic Research Archive, 2022, 30(1): 116-125. doi: 10.3934/era.2022006

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Abstract

A new bifurcation from simple eigenvalue theorem is proved for general nonlinear functional equations. It is shown that in this bifurcation scenario, the bifurcating solutions are on a curve which is tangent to the line of trivial solutions, while in typical bifurcations the curve of bifurcating solutions is transversal to the line of trivial ones. The stability of bifurcating solutions can be determined, and examples from partial differential equations are shown to demonstrate such bifurcations.

References

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