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

For citation:

Ponomarenko V. P. Dynamical modes and nonlinear phenomena in modified autooscillatory system with frequency-phase control. Izvestiya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, iss. 3, pp. 52-74. DOI: 10.18500/0869-6632-2017-25-3-52-74

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):
(downloads: 114)
Article type: 

Dynamical modes and nonlinear phenomena in modified autooscillatory system with frequency-phase control

Ponomarenko Valerij Pavlovich, Institute of Applied Mathematics and Cybernetics. Nizhny Novgorod state University

In the proposed paper, we investigate the dynamical behavior of the modified system with frequency-phase control, which uses two-channel discriminator in the circuit of phase control and multi-frequency discriminator with periodic nonlinearity in the circuit of frequency control. We consider the case of identical low-pass filters of the third order in the both control circuits. Mathematical model of analyzed frequency-phase system is presented by a nonlinear dynamical system in the four-dimensional cylindrical phase space. The model is characterized by a great number of equilibrium states. The aim of this work is to reveal new dynamical modes and nonlinear phenomena due to the specified modification. The study of the system under consideration dynamics consists in analysis of the phase synchronous mode, which is the main operating state in traditional applications of the systems with phase control, and non-synchronous modes of the frequency-phase system that serve as the basic working process in a new, non-traditional applications associated with the generation of chaotic oscillations. For solving these problems, we use computer modeling that based on qualitative and numerical methods of nonlinear dynamics. The conditions of the synchronous mode realization are determined. The existences of a great number various periodic and chaotic non-synchronous modes are established. Numerical analysis showed the existence in the system of qualitatively new quasi-synchronous and asynchronous modes, which are interesting for applied problems of generation of oscillations with chaotic modulation of their frequency and phase. In the process of numerical simulation, we have analyzed the bifurcations leading to the emergence and disappearance of non-synchronous modes. High level of the non-synchronous modes multistability of frequency-phase system is discovered. The peculiarities of the dynamics of the system caused by the parameters that characterize the inertia of the control circuits and the degree of influence of the circuit frequency control are studied. The results are presented in the form of one- and two-parameter bifurcation diagrams, phase portraits, Poincare sections and waveforms of oscillations. The revealed new non-synchronous modes of the modified system with frequency-phase control are of interest in the solution of applied problems of constructing generators of chaotic signals on the base of the systems with phase control. 

  1. Kаpranov M.V., Kuleshov V.N., Utkin G.M. Theory of Oscillations in Radio Engineering. Moscow: Nauka, 1984. 320 p. (in Russian).
  2. Shalfeev V.D., Matrosov V.V. Nonlinear Dynamics of Phase-Locked Systems. Nizhny Novgorod: Izdatel’stvo Nizhegorodskogo Universiteta, 2013. 366 p. (in Russian).
  3. Kaganov V.I. Radio Electronics Automatic Control Systems: Computerized course. Train aid for institutions of higher learning. Moscow: Hot line-Telecom, 2009. 432 p. (in Russian).
  4. Dmitriev A.S. Shirokov M.E. Choice of Generator for a Direct Chaotic Communication System. Journal of Communication Technology and Electronics. 2004. Vol. 49, No7. P. 790.
  5. Dmitriev A.S., Kletsov A.V., Kuz’min L.V. Generation of Ultrawideband Phase Chaos in the Decimeter Band. Journal of Communication Technology and Electronics. 2009. Vol. 54, No6. P. 675.
  6. Shakhtarin B.I., Kobylkina P.I., Sidorkina Yu.A., Kondrat’ev A.V., Mitin S.V. Generators of Chaotic Oscillations: Train aid. Moscow: Gelios ARV, 2007. 248 p. (in Russian).
  7. Kаpranov M.V. About the capture range at the automatic frequency-phase control. Nauchnye Doklady Vyshei Shkoly. Seriya «Radiotekhnika i Elektronika». 1958. Vol. 2, No9. P. 162. (in Russian).
  8. Kaganov V.I., Tereshenko S.V. Noise-Immunity of the Double-Loop System of Automatic Control. Journal of Communication Technology and Electronics. 2012. Vol. 57. No 3. P. 323.
  9. Ponomarenko V.P., Tikhonov E.A. Dynamics of a phase-frequency-feedback oscillator with an inverted frequency discriminator characteristic. Izvestiya VUZ. Applied Nonlinear Dynamics. 2003. Vol. 11, No6. P. 75–91. (in Russian).
  10. Ponomarenko V.P., Tikhonov E.A. Chaotic and regular dynamics of a self-oscillator system with a nonlinear frequency-phase control loop. Journal of Communication Technology and Electronics. 2004. Vol. 49, No2. P. 187.
  11. Ponomarenko V.P. Dynamical regimes in models of autooscillatory systems with frequency and frequency-phase control. Izvestiya VUZ. Applied Nonlinear Dynamics. 2007. Vol. 15, No3. P. 33–51. (in Russian).
  12. Ponomarenko V.P. Dynamical regimes and nonlinear phenomena in generator with frequency-phase control. Izvestiya VUZ. Applied Nonlinear Dynamics. 2008. Vol. 16, No 6. P. 18–40. (in Russian).
  13. Matrosov V.V. The dynamics of a frequency- and phase-controlled oscillator. Radiophysics and Quantum Electronics. 2004. Vol. 47, No 4. P. 297–304.
  14. Matrosov V.V. Modeling of dynamics of the frequency-phase control system with the first-order filters. Vestnik of Lobachevsky State University of Nizhny Novgorod. Seriya «Mathematical Modeling and Control». 2006. Vol. 2(31). P. 17–28. (in Russian).
  15. Kаpranov M.V., Romanov E.V. Linear models of the automatic frequency control system with a discriminator constructed with the use of a delay line. Radiotekhnika. 1988. No11. P. 34–38. (in Russian).
  16. Ponomarenko V.P. Nonlinear effects in autooscillatory system with frequency-phase control. Izvestiya VUZ. Applied Nonlinear Dynamics. 2012. Vol. 20, No4. P. 66–84. (in Russian).
  17. Ponomarenko V.P. Dynamic modes and bifurcations in the frequency-phase lock system with a multiple-frequency discriminator. Journal of Communication and Electronics. 2015. Vol. 60, No 2. P. 179–192.
  18. Chernobayev V.G., Kapranov M.V. Investigation of multi channel phase discriminator phase locked loops. Proceedings of 2nd International Conference «Control of Oscillations and Chaos». July 5–7, St. Petersburg, Russia. 2000. Vol. 1. P. 130–132.
  19. Chernobayev V.G., Kapranov M.V. Chaos in two-channel phase locked loops with multi-frequency discriminators. Perspective Technologies in Information. Vladimir: IENR, 1999. Vol. 2. P. 282–285.
  20. Shilnikov L.P., Shilnikov A.L., Turaev D.V., Chua L.O. Methods of Qualitative Theory in Nonlinear Dynamics. World Scientific. Singapore, New Jersey, London, Hong Kong, 2009. 548 p.
  21. Sistemu Fazovoi Sinchronizatcii. Eds V.V. Shakhgildyan, L.N.Belustina. Moscow: Radio and Svyaz, 1982. 288 p. (in Russian).
  22. Ermolayev Yu.L., Sanin A.L. Electronic Synergetics. Leningrad: Izdatel’stvo Leningradskogo Universiteta, 1989. 248 p. (in Russian).
  23. Neimark Yu.I. Dynamic Systems and Controlled Processes. Moscow: Knizhny Dom «LIBROCOM», 2014. 336 p. (in Russian).
  24. Anishchenko V.S. Complex Oscillations in the Simple Systems. Moscow: Nauka, 1990. 312 p. (in Russian).
  25. Dynamics of Nonlinear Systems. Complex of the Programs for Research of Nonlinear Dynamical Systems with Continuous Time/ Comp. by V.V. Matrosov. N. Novgorod: Izdatel’stvo Nizhegorodskogo Universiteta, 2002. 54 p. (in Russian).
  26. Bautin N.N. Behavior of the Dynamic Systems Near-by the Boundaries of Stability Domain. Moscow: Nauka, 1984. 176 p. (in Russian).
  27. R.N. Madan (Editor). Chua’s Circuits: A Paradigm for Chaos. World Scientific Series on Nonlinear Science, series B. 1993. Vol. 1. World Scientific, Singapore. 1043 p.
  28. Suykens J.A.K, Vanderwalle J. Generation of n-double scrolls (n = 1, 2, 3, 4, ...). IEEE Transaction on Circuits and Systems-1: Fundamental Theory and Applications. 1993. Vol. 40. No 11. P. 861–867.
  29. Bilotta E., Pantano P., Stranges F. A gallery of Chua attractors: Part 1. International Journal of Bifurcation and Chaos. 2007. Vol. 17, No 1. P. 1–60.
  30. Kuznetsov A.P., Sataev I.R., Stankevich N.M., Turukina L.V. Physics of Quasiperiodic Oscillations. Saratov: Nauka, 2013. 252 p. (in Russian).
  31. Dmitriev A.S., Efremova E.V., Maksimov N.A., Panas A.I. Generation of Chaos / Ed. A.S. Dmitriev. Moscow: Tekhnosfera, 2012. 424 p. (in Russian).
Short text (in English):
(downloads: 126)