Abstract
In the study of nonlinear oscillations, the question on the dependence of the frequency or the period on the energy often arises. In this paper, we find conditions under which the frequency depends on the energy monotonically. In addition, for oscillations near separatrix trajectories, an asymptotics of the period with respect to the energy is constructed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. I. Arnol’d, V. V. Kozlov, and A. I. Neishtadt, “Mathematical aspects of classical and celestial mechanics,” in: Dynamical Systems–3, Itogi Nauki Tekhn. Ser. Sovrem. Probl. Mat. Fund. Napravl., 3, All-Union Institute for Scientific and Technical Information, Moscow (1985), pp. 5–290.
M. V. Fedoryuk, Method of Steepest Descent [in Russian], Nauka, Moscow (1977).
L. A. Kalyakin, “Averaging in the autoresonance model,” Mat. Zametki, 73, No. 3, 449–452 (2003).
L. A. Kalyakin, “Asymptotic analysis of autoresonance models,” Usp. Mat. Nauk, 63, No. 5, 3–72 (2008).
Murdock J. A., Perturbations: Theory and Methods, Wiley, New York (1991).
A. I. Neishtadt, “Averaging, passage through resonances, and capture into resonance in two-frequency systems,” Uspekhi Mat. Nauk, 69, No. 5, 3–80 (2014).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 163, Differential Equations, 2019.
Rights and permissions
About this article
Cite this article
Kalyakin, L.A. On the Frequency of a Nonlinear Oscillator. J Math Sci 258, 13–22 (2021). https://doi.org/10.1007/s10958-021-05533-w
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-021-05533-w