ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Krylosova D. A., Kuznetsov A. P., Sedova Y. V., Stankevich N. V. Self-oscillating systems with controlled phase of external force. Izvestiya VUZ. Applied Nonlinear Dynamics, 2023, vol. 31, iss. 5, pp. 549-565. DOI: 10.18500/0869-6632-003057, EDN: WILGFO

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon and_2023-5_krylosova-et-al_549-565.pdf
(downloads: 0)
Full text PDF(En):
PDF icon and_2023_5_krylosova_et_al.pdf
(downloads: 4)
Language: 
Russian
Heading: 
Bifurcation in Dynamical Systems. Deterministic Chaos. Quantum Chaos.
Article type: 
Article
UDC: 
530.182
DOI: 
10.18500/0869-6632-003057
EDN: 
WILGFO

Self-oscillating systems with controlled phase of external force

Autors: 
Krylosova Darina Andreevna, Yuri Gagarin State Technical University of Saratov
Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Sedova Yuliya Viktorovna, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Stankevich Nataliya Vladimirovna, National Research University "Higher School of Economics"
Abstract: 

The purpose of this work is to study self-oscillatory systems under adaptive external action. This refers to the situation when the phase of the external action additionally depends on the dynamical variable of the oscillator. In a review plan, the results are presented for the case of a linear damped oscillator. Two cases of self-oscillatory systems are studied: the van der Pol oscillator and an autonomous quasi-periodic generator with three-dimensional phase space.

Methods. Methods of charts of dynamical regimes and charts of Lyapunov exponents are used, as well as the construction of phase portraits and stroboscopic sections.

Results. In a review plan, the results are presented for the case of a linear damped oscillator. Two cases of self-oscillatory systems are studied: the van der Pol oscillator and an autonomous quasi-periodic generator with a three-dimensional phase space. The pictures of characteristic dynamical regimes are described. Scenarios for the development of multidimensional chaos are described. Illustrations are given of the influence of the control parameter, which is responsible for the degree of dependence of the phase on the oscillator variable, on the dynamics of the system at different frequencies of action.

Conclusion. The taling into account of the dependence of the phase on a dynamical variable leads to an extension of the tongues of subharmonic resonances, which are weakly expressed in the classical van der Pol oscillator. This is especially noticeable for even resonances of periods 2 and 4. For the generator of quasi-periodic oscillations in the non-autonomous case, three-frequency tori are observed, their regions begin to dominate with an increase in the adaptivity parameter, displacing the tongues of resonant two-frequency tori. A variety of multidimensional chaos characterized by an additional Lyapunov exponent close to zero is discovered, the possibility of developing hyperchaos as a result of destruction is shown.

Key words: 
non-autonomous oscillator
phase
Van der Pol oscillator
quasi-periodicity
chaos
Acknowledgments: 
This work was supported by the Russian Science Foundation (Project no. 21-12-00121), https://rscf.ru/project/21-12-00121/
Reference: 
  1. Best RE. Phase-Locked Loops: Design, Simulation, and Applications. 6th ed. New York: McGrawHill; 2007. 489 p.
  2. Shalfeev VD, Matrosov VV. Nonlinear Dynamics of Phase Synchronization Systems. Nizhny Novgorod: Nizhny Novgorod Unoversity Publishing; 2013. 366 p. (in Russian).
  3. Kuznetsov NV, Leonov GA. Nonlinear Mathematical Models of Phase-Locked Loops. Cambridge Scientific Publisher; 2014. 218 p.
  4. Kuznetsov NV, Belyaev YV, Styazhkina AV, Tulaev AT, Yuldashev MV, Yuldashev RV. Effects of PLL architecture on MEMS gyroscope performance. Gyroscopy and Navigation. 2022;13(1):44–52. DOI: 10.1134/S2075108722010047.
  5. Kuznetsov NV, Lobachev MY, Yuldashev MV, Yuldashev RV, Tavazoei MS. The gardner problem on the lock-in range of second-order type 2 phase-locked loops. IEEE Transactions on Automatic Control. 2023:1–15. DOI: 10.1109/TAC.2023.3277896.
  6. Ottesen JT. Modelling the dynamical baroreflex-feedback control. Mathematical and Computer Modelling. 2000;31(4–5):167–173. DOI: 10.1016/S0895-7177(00)00035-2.
  7. Hall JE. Guyton and Hall Textbook of Medical Physiology E-Book. Elsevier Health Sciences; 2015. 1147 p.
  8. Seleznev EP, Stankevich NV. Complex dynamics of a non-autonomous oscillator with a controlled phase of an external force. Technical Physics Letters. 2019;45(1):57–60. DOI: 10.1134/ S1063785019010334.
  9. Krylosova DA, Seleznev EP, Stankevich NV. Dynamics of non-autonomous oscillator with a controlled phase and frequency of external forcing. Chaos, Solitons & Fractals. 2020;134:109716. DOI: 10.1016/j.chaos.2020.109716.
  10. Krylosova D, Seleznev E, Stankevich N. The simplest oscillators with adaptive properties. In: 2020 4th Scientific School on Dynamics of Complex Networks and their Application in Intellectual Robotics (DCNAIR). 07-09 September 2020, Innopolis, Russia. IEEE; 2020. P. 140–143. DOI: 10.1109/DCNAIR50402.2020.9216759.
  11. Polczynski K, Bednarek M, Awrejcewicz J. Magnetic oscillator under excitation with controlled initial phase. In: Awrejcewicz J, Kazmierczak M, Olejnik P, Mrozowski J, editors. 16th International Conference Dynamical Systems – Theory and Applications. 6-9 December 2021 Lod´z. DSTA; 2021. P. 400–401.
  12. Pikovsky A, Rosenblum M, Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences. New York: Cambridge University Press; 2001. 412 p. DOI: 10.1017/CBO9780511755743.
  13. Balanov A, Janson N, Postnov D, Sosnovtseva O. Synchronization: From Simple to Complex. Berlin: Springer; 2009. 426 p. DOI: 10.1007/978-3-540-72128-4.
  14. Landa PS. Self-Oscillations in Systems With a Finite Number of Degrees of Freedom. Moscow: Nauka; 1980. 360 p. (in Russian).
  15. Ding EJ, Hemmer PC. Winding numbers for the supercritical sine circle map. Physica D: Nonlinear Phenomena. 1988;32(1):153–160. DOI: 10.1016/0167-2789(88)90092-9.
  16. Ivankov NY, Kuznetsov SP. Complex periodic orbits, renormalization, and scaling for quasiperiodic golden-mean transition to chaos. Phys. Rev. E. 2001;63(4):046210. DOI: 10.1103/PhysRevE. 63.046210.
  17. Kuznetsov AP, Kuznetsov SP, Mosekilde E, Stankevich NV. Generators of quasiperiodic oscillations with three-dimensional phase space. The European Physical Journal Special Topics. 2013; 222(10):2391–2398. DOI: 10.1140/epjst/e2013-02023-x.
  18. Kuznetsov AP, Kuznetsov SP, Shchegoleva NA, Stankevich NV. Dynamics of coupled generators of quasiperiodic oscillations: Different types of synchronization and other phenomena. Physica D: Nonlinear Phenomena. 2019;398:1–12. DOI: 10.1016/j.physd.2019.05.014.
  19. Matsumoto T. Chaos in electronic circuits. Proceedings of the IEEE. 1987;75(8):1033–1057. DOI: 10.1109/PROC.1987.13848.
  20. Anishchenko VS, Nikolaev SM. Generator of quasi-periodic oscillations featuring two-dimensional torus doubling bifurcations. Technical Physics Letters. 2005;31(10):853–855. DOI: 10.1134/ 1.2121837.
  21. Anishchenko V, Nikolaev S, Kurths J. Winding number locking on a two-dimensional torus: Synchronization of quasiperiodic motions. Phys. Rev. E. 2006;73(5):056202. DOI: 10.1103/ PhysRevE.73.056202.
  22. Anishchenko V, Nikolaev S, Kurths J. Peculiarities of synchronization of a resonant limit cycle on a two-dimensional torus. Phys. Rev. E. 2007;76(4):046216. DOI: 10.1103/PhysRevE.76.046216.
  23. Kuznetsov AP, Kuznetsov SP, Stankevich NV. A simple autonomous quasiperiodic self-oscillator. Communications in Nonlinear Science and Numerical Simulation. 2010;15(6):1676–1681. DOI: 10.1016/j.cnsns.2009.06.027.
  24. Kuznetsov AP, Sedova YV. Anishchenko-Astakhov quasiperiodic generator excited by external harmonic force. Technical Physics Letters. 2022;48(2):84–86. DOI: 10.21883/TPL.2022.02.52858. 18925.
  25. Stankevich NV, Kuznetsov AP, Kurths J. Forced synchronization of quasiperiodic oscillations. Communications in Nonlinear Science and Numerical Simulation. 2015;20(1):316–323. DOI: 10.1016/j.cnsns.2014.04.020.
  26. Kuznetsov AP, Sataev IR, Turukina LV. Phase dynamics of periodically driven quasiperiodic self-vibrating oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(4):17–32 (in Russian). DOI: 10.18500/0869-6632-2010-18-4-17-32.
  27. Kuznetsov AP, Sataev IR, Turukina LV. Forced synchronization of two coupled van der Pol self-oscillators. Russian Journal of Nonlinear Dynamics. 2011;7(3):411–425 (in Russian). DOI: 10. 20537/nd1103001.
  28. Vitolo R, Broer H, Simo C. Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems. Regular and Chaotic Dynamics. 2011;16(1–2):154–184. DOI: 10.1134/S1560354711010060.
  29. Broer H, Simo C, Vitolo R. Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing. Nonlinearity. 2002;15(4):1205–1267. DOI: 10.1088/0951-7715/15/4/312.
  30. Broer HW, Simo C, Vitolo R. Chaos and quasi-periodicity in diffeomorphisms of the solid torus. Discrete and Continuous Dynamical Systems - B. 2010;14(3):871–905. DOI: 10.3934/dcdsb. 2010.14.871.
  31. Stankevich NV, Shchegoleva NA, Sataev IR, Kuznetsov AP. Three-dimensional torus breakdown and chaos with two zero Lyapunov exponents in coupled radio-physical generators. Journal of Computational and Nonlinear Dynamics. 2020;15(11):111001. DOI: 10.1115/1.4048025.
  32. Grines EA, Kazakov A, Sataev IR. On the origin of chaotic attractors with two zero Lyapunov exponents in a system of five biharmonically coupled phase oscillators. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2022;32(9):093105. DOI: 10.1063/5.0098163.
  33. Karatetskaia E, Shykhmamedov A, Kazakov A. Shilnikov attractors in three-dimensional orientation-reversing maps. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2021;31(1):011102. DOI: 10.1063/5.0036405.
  34. Kuznetsov AP, Sedova YV, Stankevich NV. Coupled systems with quasi-periodic and chaotic dynamics. Chaos, Solitons & Fractals. 2023;169:113278. DOI: 10.1016/j.chaos.2023.113278.
Received: 
02.06.2023
Accepted: 
10.08.2023
Available online: 
12.09.2023
Published: 
29.09.2023
  • 1251 reads