E. N. Pelinovsky^{1, 2}, I. E. Melnikov^{1, 2}

¹Higher School of Economics (Nizhny Novgorod, Russian Federation)

²Institute of Applied Physics of the Russian Academy of Sciences (Nizhny Novgorod, Russian Federation)

Abstract. Solving nonlinear differential equations with external forces is important for understanding resonant phenomena in the physics of oscillations. The article analyzes this problem basing on example of an ordinary second-order differential equation of the pendulum type, where the nonlinearity is described by a sinusoidal term. The phase plane of such an oscillator is constructed and its periodic trajectories are studied. It is illustrated that bounded nonlinearity matters only at intermediate amplitudes. The excitation of a nonlinear oscillator is carried out using a limited two–component force; the first its component corresponds to an oscillation at the resonant frequency of a linear oscillator, and the second is a limited function with a variable frequency. It is shown that with the appropriate choice of an external force, it is possible to obtain unlimited amplification of oscillations in a pendulum-type oscillator with amplitude linearly proportional to time. Spectral composition of the external force is investigated using short-time Fourier transform. It is demonstrated that in order to maintain the resonant mode, the frequency of the external force must continuously increase. Energy estimates of the external force and oscillator fluctuations depending on time are performed. The considered example is important for understanding resonant conditions in nonlinear problems.

Key Words: Nonlinear resonance, oscillator, short-time Fourier transform, mathematical pendulum, spectrogram

For citation: E. N. Pelinovsky, I. E. Melnikov. Resonance in bounded nonlinear pendulum-type systems. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 24:3(2022), 289–296. DOI: https://doi.org/10.15507/2079-6900.24.202203.289-296

Information about the authors:

Efim N. Pelinovsky, Professor of the Department of Fundamental Mathematics, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603155, Russia), Chief Researcher, Federal Research Center Institute of Applied Physics of the Russian Academy of Sciences (46 Ulyanova St., Nizhny Novgorod 603950), Doctor of Physical and Mathematical Sciences, ORCID: http://orcid.org/0000-0002-5092-0302 , pelinovsky@appl.sci-nnov.ru

Ioann E. Melnikov, Student of the Faculty of Informatics, Mathematics and Computer Science, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603150, Russia), ORCID: https://orcid.org/ 0000-0003-4560-9648, melnicovioann@gmail.com