preprint · arXiv (Cornell University) · 2025
Evans function, parity and nonautonomous bifurcations
Pötzsche, Christian, Skiba, Robert
Cited by 0Open access
Abstract
The concept of parity due to Fitzpatrick, Pejsachowicz and Rabier is a central tool in the abstract bifurcation theory of nonlinear Fredholm operators. In this paper, we relate the parity to the Evans function, which is widely used in the stability analysis for traveling wave solutions to evolutionary PDEs. As application we obtain a flexible and general condition yielding local bifurcations of specific bounded entire solutions to (Carath{é}odory) differential equations. These bifurcations are intrinsically nonautonomous in the sense that the assumptions implying them cannot be fulfilled for autonomous or periodic temporal forcings. In addition, we demonstrate that Evans functions are strictly related to the dichotomy spectrum and hyperbolicity, which play a crucial role in studying the existence of bounded solutions on the whole real line and therefore the recent field of nonautonomous bifurcation theory. Finally, by means of non-trivial examples we illustrate the applicability of our methods.
Cite this paper
Pötzsche, C., & Skiba, R. (2025). Evans function, parity and nonautonomous bifurcations. arXiv (Cornell University). https://doi.org/10.48550/arxiv.2503.07221
Pötzsche, Christian, and Robert Skiba. “Evans function, parity and nonautonomous bifurcations.” arXiv (Cornell University), 2025, https://doi.org/10.48550/arxiv.2503.07221.
Pötzsche, Christian, and Robert Skiba. 2025. “Evans function, parity and nonautonomous bifurcations.” arXiv (Cornell University). https://doi.org/10.48550/arxiv.2503.07221.
Pötzsche, C. and Skiba, R. (2025) ‘Evans function, parity and nonautonomous bifurcations’, arXiv (Cornell University). Available at: https://doi.org/10.48550/arxiv.2503.07221.
@misc{potzsche2025evans,
title = {{Evans function, parity and nonautonomous bifurcations}},
author = {Pötzsche, Christian and Skiba, Robert},
journal = {arXiv (Cornell University)},
year = {2025},
doi = {10.48550/arxiv.2503.07221}
}Made from the index record. Check names, capitalisation and page numbers against the paper. Need another style? AI for Research has over 10,000.