Nonautonomous Fold Bifurcations from Spectral Intervals of Higher Strict Multiplicity

Christian Pötzsche

doi:10.2478/tmmp-2023-0018

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Nonautonomous Fold Bifurcations from Spectral Intervals of Higher Strict Multiplicity

Tatra Mountains Mathematical Publications

Volume 84 (2023): Issue 2 (June 2023)

By: Christian Pötzsche

Open Access

|Jun 2023

Abstract

We provide two sufficient criteria for the bifurcation of bounded entire or homoclinic solutions to nonautonomous difference equations depending on a single real parameter. Our analysis is based on a nonhyperbolic solution, whose variational equation possesses exponential dichotomies on semiaxes ensuring that the corresponding critical spectral interval of the dichotomy spectrum has strict multiplicity > 1. This extends earlier results on the fold bifurcation.

References

HENRY, D.: Geometric Theory of Semilinear Parabolic Equations. In: Lect. Notes in Math. Vol. 840, Springer-Verlag, Berlin, 1981.
Search in Google Scholar Back to article
LANG, S: Real and Functional Analysis. 3rd edition. In: Graduate Texts in Mathematics Vol. 142, Springer-Verlag, 1993.
Search in Google Scholar Back to article
LIU, P.—WANG, Y.: The generalized saddle-node bifurcation of degenerate solution. Ann. Soc. Math. Pol., Ser. I, Commentat. Math 45 (2005), no. 2, 145–150.
Search in Google Scholar Back to article
LIU, P.—WANG, Y.: A fold bifurcation theorem of degenerate solutions in a perturbed nonlinear equation. Abst. Appl. Anal. 2011 (2011), Article ID 302942, 11 pp.
Search in Google Scholar Back to article
LIU, P.—SHI, B.—WANG, Y.: A double saddle-node bifurcation theorem. Commun. Pure Appl. Anal. 12 (2013), no. 6, 2923–2993.
Search in Google Scholar Back to article
PÖTZSCHE, C.: Nonautonomous continuation of bounded solutions, Commun. Pure Appl. Anal. 10 (2011), no. 3, 937–961.
Search in Google Scholar Back to article
PÖTZSCHE, C.: Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), no. 2, 739–776.
Search in Google Scholar Back to article
RUSS, E.: Dichotomy spectrum for difference equations in Banach spaces, J. Difference Equ. Appl. 23 (2016), no. 3, 576–617.
Search in Google Scholar Back to article
ZEIDLER, E.: Nonlinear functional analysis and its applications I (Fixed-points theorems) Springer-Verlag, Berlin, 1993.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/tmmp-2023-0018 | Journal eISSN: 1338-9750 | Journal ISSN: 1210-3195

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Language: English

Page range: 87 - 110

Submitted on: Nov 7, 2022

Published on: Jun 28, 2023

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

nonautonomous bifurcation,

nonautonomous difference equation,

Related subjects:

© 2023 Christian Pötzsche, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 84 (2023): Issue 2 (June 2023)