DYNAMICS OF COUPLED GENERATORS OF QUASI-PERIODIC OSCILLATIONS WITH EQUILIBRIUM STATE
Cite this article as:
Kuznetsov A. P., Stankevich N. V. DYNAMICS OF COUPLED GENERATORS OF QUASI-PERIODIC OSCILLATIONS WITH EQUILIBRIUM STATE. Izvestiya VUZ. Applied Nonlinear Dynamics, 2018, vol. 26, iss. 2, pp. 41-58. DOI: https://doi.org/10.18500/0869-6632-2018-26-2-41-58
Subject of the study. Recently, the problems of synchronization of systems demonstrating
quasi-periodic oscillations arouse interest. In particular, it can be generators of quasi-periodic oscillations that allow a radiophysical realization. In this paper we consider the dynamics of two coupled oscillators of quasi-periodic oscillations with a single equilibrium state. Novelty. The difference from the already studied case of coupled modified Anishchenko–Astakhov generators consists in engaging of two-parameter analysis and analysis in a much wider range of parameter changes, as well as a more dimensionless equation for an individual generator. Methods. The method of charts of Lyapunov exponents is used, which reveals areas of various types of dynamics, up to four-frequency oscillations. The bifurcation mechanisms of complete synchronization are investigated. Results. The possibility of synchronous quasi- periodicity is demonstrated, when the phases of the generators are locked, but the dynamics of the system is generally quasi-periodic. The possibility of the effect of «death of oscillations» arising due to the dissipative character of coupling is revealed. The possibility of the effect of broadband quasi-periodicity is demonstrated. Its peculiarity consists in the fact that two-frequency oscillations arise in a certain range of variation of the coupling parameter and a wide range of frequency mismatch. The bifurcation mechanisms of this effect are presented. It is shown that a certain degeneracy is characteristic for it, which is removed when nonidentity is introduced along the control parameters of individual generators. A bifurcation analysis is presented for this case. Two-parameter analysis allowed us to identify points of quasi-periodic bifurcations of codimension two QSNF (Quasi-periodic saddle-node fan) on the parameter plane, associated with the synchronization of multi-frequency tori. These points are the tips of the tongues of the two-frequency regimes, which have a threshold for the coupling coefficient. In their vicinity, three- and four-frequency quasi-periodic regimes are also observed. Discussion. Synchronization of quasi-periodic generators has a number of new moments that are established in two-parameter analysis in a wide range of parametric changes.
DOI: 10.18500/0869-6632-2018-26-2-41-58
References: Kuznetsov A.P., Stankevich N.V. Dynamics of coupled generators of quasi-periodic oscillations with equilibrium state. Izvestiya VUZ, Applied Nonlinear Dynamics, 2018, vol. 26, iss. 2, pp. 41–58. DOI: 10.18500/0869-6632-2018-26-2-41-58
BibTeX
author = {A. P. Kuznetsov and Nataliya Vladimirovna Stankevich},
title = {DYNAMICS OF COUPLED GENERATORS OF QUASI-PERIODIC OSCILLATIONS WITH EQUILIBRIUM STATE},
year = {2018},
journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
volume = {26},number = {2},
url = {https://old-andjournal.sgu.ru/en/articles/dynamics-of-coupled-generators-of-quasi-periodic-oscillations-with-equilibrium-state},
address = {Саратов},
language = {russian},
doi = {10.18500/0869-6632-2018-26-2-41-58},pages = {41--58},issn = {0869-6632},
keywords = {quasi-periodic oscillations,coupled generators,synchronization},
abstract = {Subject of the study. Recently, the problems of synchronization of systems demonstrating quasi-periodic oscillations arouse interest. In particular, it can be generators of quasi-periodic oscillations that allow a radiophysical realization. In this paper we consider the dynamics of two coupled oscillators of quasi-periodic oscillations with a single equilibrium state. Novelty. The difference from the already studied case of coupled modified Anishchenko–Astakhov generators consists in engaging of two-parameter analysis and analysis in a much wider range of parameter changes, as well as a more dimensionless equation for an individual generator. Methods. The method of charts of Lyapunov exponents is used, which reveals areas of various types of dynamics, up to four-frequency oscillations. The bifurcation mechanisms of complete synchronization are investigated. Results. The possibility of synchronous quasi- periodicity is demonstrated, when the phases of the generators are locked, but the dynamics of the system is generally quasi-periodic. The possibility of the effect of «death of oscillations» arising due to the dissipative character of coupling is revealed. The possibility of the effect of broadband quasi-periodicity is demonstrated. Its peculiarity consists in the fact that two-frequency oscillations arise in a certain range of variation of the coupling parameter and a wide range of frequency mismatch. The bifurcation mechanisms of this effect are presented. It is shown that a certain degeneracy is characteristic for it, which is removed when nonidentity is introduced along the control parameters of individual generators. A bifurcation analysis is presented for this case. Two-parameter analysis allowed us to identify points of quasi-periodic bifurcations of codimension two QSNF (Quasi-periodic saddle-node fan) on the parameter plane, associated with the synchronization of multi-frequency tori. These points are the tips of the tongues of the two-frequency regimes, which have a threshold for the coupling coefficient. In their vicinity, three- and four-frequency quasi-periodic regimes are also observed. Discussion. Synchronization of quasi-periodic generators has a number of new moments that are established in two-parameter analysis in a wide range of parametric changes. DOI: 10.18500/0869-6632-2018-26-2-41-58 References: Kuznetsov A.P., Stankevich N.V. Dynamics of coupled generators of quasi-periodic oscillations with equilibrium state. Izvestiya VUZ, Applied Nonlinear Dynamics, 2018, vol. 26, iss. 2, pp. 41–58. DOI: 10.18500/0869-6632-2018-26-2-41-58 }}