Averaging methods for nonlinear systems with a small parameter via reduction and topological degree. (February 2019)

• Record Type: Journal Article
• Authors: Liu, Qihuai; Cai, Luocheng
• Is Part Of: Nonlinear analysis. Volume 45(2019)
• Page Start: 461
• Page End: 471
• Publication Date: 2019-02
• Subjects: Periodic solution -- Lyapunov–Schmidt reduction -- Averaging method -- Topological degree; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2018.07.019
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: When using averaging theory, one usually tries to find the simple zeros of an associated averaged function. When the zero is not simple, the classical averaging theory does not provide information about the periodic solution associated to a non-simple zero. To overcome this difficulty, one way is to perform the high order averaging in case that the system is smooth enough. Here we provide sufficient conditions to keep periodic solutions persisting by the Lyapunov–Schmidt reduction and topological degree theory. Additionally, we perform an application of our result for the forced Galactic dynamics model. Highlights: We study the continuation of periodic solutions with non-simple zeros. Averaging is constructed by the Lyapunov–Schmidt reduction and topological theory. Periodic solutions of the forced Galactic dynamics model are obtained.