# Doubochinski's pendulum

Doubochinski's pendulum is a classical oscillator interacting with a high-frequency field in such a way that the oscillator takes on a discrete set of stable regimes of oscillation, each at a frequency near to the proper frequency of the oscillator, but each with a distinct, "quantized" amplitude.[1][2][3][4][5][6] The phenomenon of amplitude quantization in this sort of coupled system was first discovered by the brothers Danil and Yakov Doubochinski in 196869.

A simple demonstration apparatus consists of a mechanical pendulum interacting with a magnetic field.[7][8] The system is composed of two interacting oscillatory processes: a pendulum arm with a natural frequency on the order of 0.51 Hz, with a small permanent magnet fixed at its moving end; and a stationary electromagnet (solenoid) positioned under the equilibrium point of the pendulum's trajectory and supplied with alternating current of fixed frequency, typically in the range of 101000 Hz.

The mechanical pendulum arm and solenoid are configured in such a way, that the pendulum arm interacts with the oscillating magnetic field of the solenoid only over a limited portion of its trajectory the so-called "zone of interaction" outside of which the strength of the magnetic field drops off rapidly to zero. This spatial inhomogeniety of the interaction is key to the discretized oscillation amplitudes and other unusual properties of the system.

## References

1. L. A. Vainshtein; Ya. B. Doubochinski (1978). "On the low-frequency oscillations under the influence of high-frequency force". Zh. Tekh. Fiz [Sov. Phys.-Tech. Phys]. 48. [23] (1494): 1321 [745].
2. D. B. Doubochinski; Ya. B. Duboshinsky; et al. (1979). "Discrete modes of a system subject to an inhomogeneous high-frequency force". Zh. Tech. Fiz [Sov. Phys.-Tech. Phys]. 49. [24]: 1160 [642].
3. P. S. Landa (2001). Nonlinear Oscillations and Waves in Dynamical Systems (PDF). Kluwer Academic Publishers. p. 307.
4. Weldon J. Wilson (2010). "Amplitude Quantization as a Fundamental Property of Coupled Oscillator Systems". Professor of Engineering and Physics, Weldon Wilson's Home Page.
5. Luo, Yao; Fan, Wenkai; Feng, Chenghao; Wang, Sihui; Wang, Yinlong (2020). "Subharmonic frequency response in a magnetic pendulum". American Journal of Physics. 88 (2): 115–123. doi:10.1119/10.0000038. We study the subharmonic frequency response of a generalized driven oscillator excited by a nonlinear periodic force. We take a magnetic pendulum called the Doubochinski pendulum as an example.
6. Pedersen, Henrik B; Madsen, Magnus Linnet; Andersen, John E V; Nielsen, Torsten G (10 February 2021). "Investigation of argumental oscillations of a physical pendulum". European Journal of Physics. 42 (2): 025012. doi:10.1088/1361-6404/abcee4. For one realization of argumental oscillations (Doubochinski's pendulum), we report experimental frequency response curves
7. D. B. Doubochinski; Ya. B. Doubochinski (1982). "Wave excitation of an oscillator having a discrete series of stable amplitudes". Dokl. Akad. Nauk SSSR [Sov. Phys. Doklady]. 3. [27]: 605 [564]. Bibcode:1982SPhD...27..564D.
8. Martin Beech (2014). The Pendulum Paradigm: Variations on a Theme and the Measure of Heaven and Earth. Universal Publishers. p. 232. ISBN 9781612337302.