Mechanisms for the origin of hidden synchronization of dynamic systems

S. S. Mamonov ; I. V. Ionova ; A. O. Kharlamova

Čebyševskij sbornik, Tome 20 (2019) no. 3, pp. 333-348.

Résumé

One of the varieties of radio engineering systems is considered in the work, namely, a frequency-phase locked loop system FPLL. The mathematical model of such a system is described by a system of differential equations with a cylindrical phase space. For the FPLL system, the conditions for the formation of latent synchronization modes are defined. Despite the numerous works devoted to FPLL systems, the questions of finding hidden synchronization, determining the mechanisms of its occurrence, finding the conditions of bifurcation of cycles and studying their scenarios, the occurrence of complex modulated oscillations remain open. The conditions for the formation of hidden synchronization are the presence in the phase-locked loop system of the frequency of the beating modes, vibrational-rotational cycles, and the presence of multistability. By multistability we understand the coexistence of several attractors in the phase space, in particular, limit cycles can be attractors. One of the cases of multistability is phase multistability, when the attractors differ from each other by the values of the phase difference between the oscillations of the system. The phase space in systems with phase multistability is more complicated than in systems with a single stable limit cycle. In the formation of multistability, the decisive role is played by unstable limit sets corresponding to oscillations not observed in the experiment. In this regard, the development of methods for determining multistability and determining the mechanisms of its appearance is relevant.In connection with the above, the urgent task is to develop numerical algorithms that allow one to find complex modulated oscillations in radio engineering systems and determine the mechanisms of their occurrence.Analytical methods for determining the latent synchronization of the PLL system are proposed, which allow developing effective computational methods for studying mathematical models of radio engineering systems using computer technologies.

Détail

Keywords: hidden synchronization, frequency-phase locked loop, quasi-synchronous mode, beat mode, vibrational cycles, rotational cycles, positively invariant set, vector field rotation, multipliers, curvature, Lyapunov exponents.

Bibliographie

[1] Bakunov G. M., Matrosov V. V., Shalfeev V. D., “On quasisynchronous modes in a phase-locked loop with a second-order filter with approximate consideration of delay”, Izv. vuzov “PND”, 19:3 (2011), 171–178 | Zbl

[2] Leonov G. A., Burkin I. M., Shepelyavy A. I., Frequency methods in the theory of oscillations, Science Publ. of St. Petersburg State University, SPb., 1992 | MR

[3] Mamonov S. S., “Dynamics of a frequency-phase locked loop with first-order filters”, Bulletin of the Novosib. State University. Series: Mathematics, Mechanics, Computer Science, 11:1 (2011), 70–81 | Zbl

[4] Mamonov S. S., Ionova I. V., “Application of rotation of a vector field to determine cycles of the second kind”, Bulletin of the Russian Academy of Natural Sciences. Differential equations, 14:5 (2014), 46–54

[5] Mamonov S. S., Kharlamova A. O., “Quasisynchronous regimes of the phase system”, Bulletin of the Ryazan State Radio Engineering University, 2016, no. 56, 45–51

[6] Mamonov S. S., Kharlamova A. O., “Separation of cycles of the second kind of frequency-phase locked loop”, Bulletin of the Russian Academy of Natural Sciences. Differential equations, 15:3 (2015), 97–102

[7] Mamonov S. S., Kharlamova A. O., “Determination of the conditions for the existence of limit cycles of the first kind of systems with cylindrical phase space”, Journal of the Srednevolzhsky Mathematical Society, 19:1 (2017), 67–76 | MR | Zbl

[8] Mamonov S. S., Kharlamova A. O., “Forced synchronization of phase-locked loop systems with delay”, Bulletin of the Ryazan State Radio Engineering University, 2017, no. 62, 26–35

[9] Mamonov S. S., Kharlamova A. O., “Numerical and analytical definition of cycles of the first kind of a phase system of differential equations”, Bulletin of the Russian Academy of Natural Sciences. Differential equations, 17:4 (2017), 48–56 | MR

[10] Mamonov S. S., Kharlamova A. O., “Cycles of the first kind of systems with a cylindrical phase space”, Results of science and technology. Series: Contemporary Mathematics and its Applications, 148, no. 4, VINITI, 2018, 83–92 | MR

[11] Mamonov S. S., Kharlamova A. O., Ionova I. V., “Oscillatory-rotational cycles of the phase system of differential equations”, Bulletin of the Russian Academy of Natural Sciences. Differential equations, 18:4 (2018), 51–57

[12] Mamonov S. S., Kharlamova A. O., Ionova I. V., “The curvature of the oscillatory cycles of phase systems”, Bulletin of the Russian Academy of Natural Sciences. Differential equations, 19:2 (2019), 105–110

[13] Shalfeev V. D., Matrosov V. V., Nonlinear dynamics of phase synchronization systems, Publ. of the UNN, N. Novgorod, 2013

[14] Matrosov V., Forced synchronization, Textbook. Allowance, Nizhny Novgorod, 2013

[15] Bakunov G. M., Matrosov V. V., Shalfeev V. D., “About regular quasysynchronous modes in the phase-locked loop”, Bulletin of Nizhny Novgorod State University, 2010, no. 6, 43–47

[16] Abramov V. V., “Conditions for the existence of a periodic solution of a second-order differential equation with a quadratic nonlinear part”, Bulletin of the Russian Academy of Natural Sciences. Differential equations, 18:4 (2018), 3–7 | Zbl

[17] Liskina E. Yu., “Conditions for the existence of nonzero periodic solutions of a second-order nonlinear autonomous dynamical system in the case of a pair of zero eigenvalues of the matrix of the linear approximation system”, Bulletin of the Russian Academy of Natural Sciences. Differential equations, 17:4 (2017), 38–43

[18] Besekersky V. A., Popov E. P., Theory of automatic control systems, Nauka, M., 1972

[19] Kapranov M. V., Kuleshov V. N., Utkin G. M., The theory of oscillations in radio engineering, Nauka, M., 1984

[20] Krasnoselsky M. A., The shift operator along the trajectories of differential equations, Nauka, M., 1966 | MR