On the Phenomenon of Low-Frequency, Large-Amplitude Oscillations in a High-Dimensional Linear Dynamical System

A. I. Gudimenko ; A. V. Lihosherstov

Russian journal of nonlinear dynamics, Tome 20 (2024) no. 2, pp. 259-276.

Résumé

This article considers a linear dynamic system that models a chain of coupled harmonic oscillators, under special boundary conditions that ensure a balanced energy flow from one end of the chain to the other. The energy conductivity of the chain is controlled by the parameter $\alpha$ of the system. In a numerical experiment on this system, with a large number of oscillators and at certain values of $\alpha$, the phenomenon of low-frequency high-amplitude oscillations was discovered. The primary analysis showed that this phenomenon has much in common with self-oscillations in nonlinear systems. In both cases, periodic motion is created and maintained by an internal energy source that does not have the corresponding periodicity. In addition, the amplitude of the oscillations significantly exceeds the initial state amplitude. However, this phenomenon also has a fundamental difference from self-oscillations in that it is controlled by the oscillation synchronization mechanism in linear systems and not by the exponential instability suppression mechanism in nonlinear systems. This article provides an explanation of the observed phenomenon on the basis of a complete analytical solution of the system. The solution is constructed in a standard way by reducing the dynamic problem to the problem of eigenvalues and eigenvectors for the system matrix. When solving, we use methods from the theory of orthogonal polynomials. In addition, we discuss two physical interpretations of the system. The connection between these interpretations and the system is established through the Schrödinger variables.

Détail

Keywords: linear dynamical system
Mots-clés : harmonic chain, high-amplitude oscillations

Bibliographie

[1] Müller, P. C. and Schiehlen, W. O., Linear Vibrations: A Theoretical Treatment of Multi-Degree-of-Freedom Vibrating Systems, translated from the German by S. Swierczkowski, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics. Dynamical Systems, 7, Martinus Nijhoff Publ., The Hague, 1985, x, 327 pp. | MR | Zbl

[2] Gatti, P. L., Applied Structural and Mechanical Vibrations: Theory and Methods, 2nd ed., CRC, London, 2014, 668 pp.

[3] Pikovsky, A., Rosenblum, M., and Kurths, J., Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Sci. Ser., 12, Cambridge Univ. Press, New York, 2001, 432 pp. | MR | Zbl

[4] Andronov, A. A., Vitt, A. A., and Khaikin, S. E., Theory of Oscillators, Pergamon, Oxford, 1966, 848 pp. | MR | Zbl

[5] Jenkins, A., “Self-Oscillation”, Phys. Rep., 525:2 (2013), 167–222 | DOI | MR | Zbl

[6] Monkewitz, P. A., Bechert, D. W., Barsikow, B., and Lehmann, B., “Self-Excited Oscillations and Mixing in a Heated Round Jet”, J. Fluid Mech., 213:1 (1990), 611–639 | DOI

[7] Zheng, Z. and Qian, Y., “Dominant Phase-Advanced Driving Analysis of Self-Sustained Oscillations in Biological Networks”, Chin. Phys. B, 27:1 (2018), 018901, 10 pp. | DOI | MR

[8] Li, M. and Bao, J., “Effect of Self-Oscillation on Escape Dynamics of Classical and Quantum Open Systems”, Entropy, 22:8 (2020), Paper No. 839, 10 pp. | MR

[9] Arnaut, L. G. and Ibánez, S., “Self-Sustained Oscillations and Global Climate Changes”, Sci. Rep., 10:1 (2020), Art. 11200, 19 pp. | DOI

[10] Zhang, H. and Detournay, E., “An Alternative Formulation for Modeling Self-Excited Oscillations of Rotary Drilling Systems”, J. Sound Vibration, 474 (2020), Art. 115241, 22 pp. | DOI | MR

[11] Zhai, C. and Sun, W., “Analytical Approximation of a Self-Oscillatory Reaction System Using the Laplace – Borel Transform”, Chaos Solitons Fractals, 142 (2021), Paper No. 110508, 12 pp. | DOI | MR

[12] Balmashnov, A. A., Kalashnikov, A. V., Kalashnikov, V. V., Stepina, S. P., and Umnov, A. M., “Self-Excitation of Low-Frequency Oscillations in the Plasma Ring Formed by an ECR Discharge in a Narrow Coaxial Cavity”, Plasma Phys. Rep., 44:6 (2018), 626–629 | DOI

[13] Chicone, C., Ordinary Differential Equations with Applications, Texts Appl. Math., 34, 2nd ed., Springer, New York, 2006, xx, 636 pp. | MR | Zbl

[14] Teschl, G., Ordinary Differential Equations and Dynamical Systems, Grad. Stud. Math., 140, AMS, Providence, R.I., 2011, 356 pp. | MR

[15] Schrödinger, E., “Zur Dynamik elastisch gekoppelter Punktsysteme”, Ann. Phys., 44 (1914), 916–934 | DOI

[16] Takizawa, É. I. and Kobayasi, K., “Heat Flow in a System of Coupled Harmonic Oscillators”, Chinese J. Phys., 1:2 (1963), 59–73

[17] Kobayasi, K. and Takizawa, É. I., “Effect of a Light Isotopic Impurity on the Energy Flow in a System of One-Dimensional Coupled Harmonic Oscillators”, Chinese J. Phys., 2:2 (1964), 68–79

[18] Olson, H. F., Dynamical Analogies, D. Van Nostrand, New York, 1943, 208 pp.

[19] Nilsson, J. W. and Riedel, S. A., Electric Circuits, 12th ed., Pearson, New York, 2023, 817 pp.

[20] Groszkowski, J., Frequency of Self-Oscillation, Pergamon, Oxford, 2013, 542 pp.

[21] Pozar, D. M., Microwave Engineering, Wiley, New York, 2011, 752 pp.

[22] Ulaby, F. T. and Ravaioli, U., Fundamentals of Applied Electromagnetics, Pearson, New York, 2023, 496 pp.

[23] Irwin, J. D. and Nelms, R. M., Basic Engineering Circuit Analysis, 12th ed., Wiley, New York, 2020, 768 pp.

[24] Lee, J. M., Manifolds and Differential Geometry, Grad. Stud. in Math., 107, AMS, Providence, R.I., 2009, xiv, 671 pp. | DOI | MR | Zbl

[25] Gantmacher, F. R., The Theory of Matrices: Vol. 1, Chelsea, New York, 1998, x, 374 pp. | MR | Zbl

[26] Gantmacher, F. P. and Krein, M. G., Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, rev. ed., translation based on the 1941 Russian original, edited and with a preface by Alex Eremenko, AMS, Providence, R.I., 2002, viii, 310 pp. | MR | Zbl

[27] Atkinson, F. V., Discrete and Continuous Boundary Problems, Math. Sci. Eng., 8, Acad. Press, New York, 1964, xiv, 570 pp. | MR | Zbl

[28] Guseinov, G. Sh., “On an Inverse Problem for Two Spectra of Finite Jacobi Matrices”, Appl. Math. Comput., 218:14 (2012), 7573–7589 | MR | Zbl

[29] Akhiezer, N. I., The Classical Moment Problem and Some Related Questions in Analysis, Oliver Boyd, Edinburgh, 1965, 253 pp. | MR | Zbl

[30] Elaydi, S., An Introduction to Difference Equations, Springer, New York, 2005, xxii, 540 pp. | MR | Zbl