Abstract
Consideration was given to an autonomous model containing coupled subsystems (MCCS) and, in the absence of couplings between the subsystems, falling down into systems of ordinary differential equations. It is assumed that the subsystems admit different types of nondegenerate single-frequency oscillations. Solved were the problems of oscillations in MCCS, their stability, and stabilization of MCCS oscillation by smooth autonomous coupling controls. It was shown how the developed theory is applied to the coupled Duffing and Van der Pol oscillators.
Similar content being viewed by others
References
Rompala, K., Rand, R., and Howland, H., Dynamics of Three Coupled Van der Pol Oscillators with Application to Circadian Rhythms, Commun. Nonlin. Sci. Numer. Simul., 2007, vol. 12, no. 5, pp. 794–803.
Yakushevich, L.V., Gapa, S., and Awrejcewicz, J., Mechanical Analog of the DNA Base Pair Oscillations, in 10th Conf. on Dynamical Systems Theory and Applications, Lodz: Left Grupa, 2009, pp. 879–886.
Kondrashov, R.E. and Morozov, A.D., On Studying Resonances in the System of two Duffing-Van der pol Equations, Nelin. Dinam., 2010, vol. 6, no. 2, pp. 241–254.
Danzl, P. and Moehlis, J., Weakly Coupled Parametrically Forced Oscillator Networks: Existence, Stability, and Symmetry of Solutions, Nonlin. Dynam., 2010, vol. 59, no. 4, pp. 661–680.
Kuznetsov, A.P., Sataev, I.R., and Tyuryukina, L.V., Forced Synchronization of two Coupled Autooscillation Van der Pol Oscillators, Nelin. Dinam., 2011, vol. 7, no. 3, pp. 411–425.
Lazarus, L. and Rand, R.H., Dynamics of a System of Two Coupled Oscillators which are Driven by a Third Oscillator, J. Appl. Nonlin. Dynam., 2014, vol. 3, no. 3, pp. 271–282.
Kawamura, Y., Collective Phase Dynamics of Globally Coupled Oscillators: Noise-induced Anti-phase Synchronization, Phys. D: Nonlin. Phenom., 2014, vol. 270, no. 1, pp. 20–29.
Peng Du and Michael Y.Li., Impact of Network Connectivity on the Synchronization and Global Dynamics of Coupled Systems of Differential Equations, Phys. D: Nonlin. Phenom., 2014, vols. 286–287, pp. 32–42.
Buono, P.-L., Chan, B.S., Palacios, A., et al., Dynamics and Bifurcations in a Dn-symmetric Hamiltonian Network. Application to Coupled Gyroscopes, Phys. D: Nonlin. Phenom., 2015, vol. 290, no. 1, pp. 8–23.
Tkhai, V.N., Model with Coupled Subsystems, Autom. Remote Control, 2013, vol. 74, no. 6, pp. 919–931.
Matrosov, V.M., The Method of Vector Lyapunov Functions in Analysis of Composite Systems with Distributed Parameters (Survey), Autom. Remote Control, 1973, vol. 34, no. 1, pp. 1–15.
Metod vektornykh funktsii Lyapunova v teorii ustoichivosti (Method of Lyapunov Vector Functions in the Stability Theory), Voronov, A.A. and Matrosov, V.M., Eds., Moscow: Nauka, 1987.
Matrosov, V.M., Metod vektornykh funktsii Lyapunova: analiz dinamicheskikh svoistv nelineinykh sistem (Method of Lyapunov Vector Functions: Analysis of the Dynamic Properties of Nonlinear Systems), Moscow: Fizmatlit, 2001.
Merkin, D.R., Giroskopicheskie sistemy (Gyro Systems), Moscow: Nauka, 1974.
Zubov, V.I., Analiticheskaya dinamika sistemy tel (Analytical Dynamics of a Body System), Leningrad: Leningr. Gos. Univ., 1983.
Pyatnitskii, E.S., Principle of Decomposition in the Control of Mechanical Systems, Dokl. Akad. Nauk SSSR, 1988, vol. 300, no. 2, pp. 300–303.
Siljak, D.D., Decentralized Control of Complex Systems, Cambridge: Academic, 1991. Translated under the title Detsentralizovannoe upravlenie slozhnymi sistemami, Moscow: Mir, 1994.
Chernous’ko, F.L., Anan’evskii, I.M., and Reshmin, S.A., Metody upravleniya nelineinymi mekhanicheskimi sistemami (Methods to Control Nonlinear Mechanical Systems), Moscow: Fizmatlit, 2006.
Barabanov, I.N., Tureshbaev, A.T., and Tkhai, V.N., Basic Oscillation Mode in the Coupled-Subsystems Model, Autom. Remote Control, 2014, vol. 75, no. 12, pp. 2112–2123.
Tkhai, V.N., Oscillations in the Autonomous Model Containing Coupled Subsystems, Autom. Remote Control, 2015, vol. 76, no. 1, pp. 64–71.
Tkhai, V.N., Oscillations,Stability and Stabilization in the Model Containing Coupled Subsystems with Cycles, Autom. Remote Control, 2015, vol. 76, no. 7, pp. 1169–1178.
Tkhai, V.N., A Mechanical System Containing Weakly Coupled Subsystems, J. Appl. Math. Mech., 2013, vol. 77, no. 6, pp. 588–594.
Tkhai, V.N., Periodic Motions of the Perturbed Reversible Mechanical System, J. Appl. Math. Mech., 2015, vol. 79, no. 2, pp. 122–131.
Malkin, I.G., Teoriya ustoichivosti dvizheniya (Theory of Motion Stability), Moscow: Nauka, 1966.
Barabanov, I.N. and Tkhai, V.N., Oscillation Family in Weakly Coupled Identical Systems, Autom. Remote Control, 2016, vol. 77, no. 4, pp. 561–569.
Malkin, I.G., Nekotorye zadachi teorii nelineinykh kolebanii (Some Problems of the Theory of Nonlinear Oscillations), Moscow: Gostekhizdat, 1956.
Andronov, A.A. and Vitt, A.A., On Laypunov Stability, Zh. Eksp. Teor. Fiz., 1933, vol. 3, no. 5, pp. 373–374.
Tkhai, V.N., Invertible Mechanical Systems Admitting Two Fixed Sets, in Problemy analiticheskoi mekhaniki i teorii ustoichivosti. Sb. nauchnykh statei, posvyashchennykh pamyati V.V. Rymyantseva (Problems of Analytical Mechanics and Theory of Stability, Collected Papers in Memoriam of V.V. Rumyantsev), Moscow: Fizmatlit, 2009, pp. 176–189.
Tkhai, V.N., On Behavior of the Period of Symmetrical Periodic Motions, Prikl. Mat. Mekh., 2012, vol. 76, no. 4, pp. 616–622.
Tkhai, V.N., On the Lyapunov–Poincare Method in the Theory of Periodic Motions, Prikl. Mat. Mekh., 1998, vol. 62, no. 3, pp. 355–371.
Coron, J.M., On the Stabilization of Controllable and Observable Systems by an Output Feedback Law, Math. Control Signal. Syst., 1994, no. 7, pp. 187–216.
Tkhai, V.N., Stabilizing the Oscillations of an Autonomous System, Autom. Remote Control, 2016, vol. 77, no. 6, pp. 972–980.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.N. Tkhai, 2017, published in Avtomatika i Telemekhanika, 2017, No. 4, pp. 21–36.
This paper was recommended for publication by L.B. Rapoport, a member of the Editorial Board
Rights and permissions
About this article
Cite this article
Tkhai, V.N. Model containing coupled subsystems with oscillations of different types. Autom Remote Control 78, 595–607 (2017). https://doi.org/10.1134/S0005117917040026
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117917040026