Abstract
In these lecture notes, we present a basic mathematical theory of parametric resonance. This includes the Floquet stability theory for periodic systems of linear differential equations and its implications for nonlinear stability. As applications to mechanical systems, we consider a pendulum with a moving support. In the end we discuss other applications in physics. We refer the reader for further studies to the books [1, 2, 6].
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References
V. I. Arnol’d. Ordinary differential equations. Springer, 1992.
V. I. Arnol’d. Mathematical methods of classical mechanics. Springer, 2013.
P. D. Lax. Linear algebra and its applications. Wiley, 2007.
A. Markeev. Stability of an equilibrium position of a pendulum with step parameters. International Journal of Non-Linear Mechanics, 73:12–17, 2015.
W. Paul. Electromagnetic traps for charged and neutral particles. Reviews of Modern Physics, 62(3):531, 1990.
A. P. Seyranian and A. A. Mailybaev. Multiparameter stability theory with mechanical applications. World Scientific, 2003.
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Mailybaev, A.A. (2024). Brief Introduction to the Theory of Parametric Resonance. In: Castilho Piqueira, J.R., Nigro Mazzilli, C.E., Pesce, C.P., Franzini, G.R. (eds) Lectures on Nonlinear Dynamics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-031-45101-0_1
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DOI: https://doi.org/10.1007/978-3-031-45101-0_1
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