The mappings of van der Pol–Dyuffing generator in discrete time

V. V. Zaitsev ; A. N. Shilin

Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 2 (2017), pp. 51-59

Résumé

In the work transition to discrete time in the equation of movement of van der Pol–Dyuffing generator is described. The transition purpose — to create mappings of the generator as subjects of the theory of nonlinear oscillations (nonlinear dynamics) in discrete time. The method of sampling is based on the use of counting of the pulse characteristic of an oscillatory contour as the sampling series for a signal in a self-oscillating ring “active nonlinearity–the resonator–feedback”. The choice of the consecutive scheme of excitement of a contour allows to receive the iterated displays in the form of recurrent formulas. Two equivalent forms of discrete displays of the generator of van der Pol–Dyuffing — complex and valid are presented. In approximation of method of slow-changing amplitudes it is confirmed that the created discrete mappings have dynamic properties of an analog prototype. Also within the numerical experiment it is shown that in case of the high power of generation the effect of changing of frequencies of harmonicas of the generated discrete signal significantly influence dynamics of the self-oscillators. In particular, in the discrete generator of van der Pol–Dyuffing the chaotic self-oscillations are observed.

Détail

Keywords: self-oscillatory system, pulse characteristic, discrete mapping, method of slow-changing amplitudes, chaotic self-oscillations.

Bibliographie

[1] Kuznetsov A. P., Kuznetsov S. P., Ryskin N. M., Nonlinear oscillations, Fizmatlit, M., 2005, 292 pp. (in Russian)

[2] Kuznetsov A. P., Seliverstova E. S., Trubetskov D. I., Turukina L. V., “Phenomenon of van der Pol equation”, Izvestiya VUZ. Applied Nonlinear Dynamics, 22:4 (2014), 3–42 (in Russian)

[3] Kalyanov E. V., Kislov V. Ya., “Self-oscillatory systems with chaotic dynamics on the basis of van der Pol –Dyuffing equations”, Journal of Communications Technology and Electronics, 51:1 (2006), 65–73 (in Russian)

[4] Kuznetsov A. P., Stankevich N. V., Turukina L. V., “Coupled van der Pol and van der Pol–Duffing oscillators: dynamics of phase and computer simulation”, Izvestiya VUZ. Applied Nonlinear Dynamics, 1:4 (2008), 101–136 (in Russian)

[5] Oppenheim A., Schafer R., Digital signal processing, Tekhnosfera, M., 2006, 856 pp. (in Russian)

[6] Zaslavsky G. M., Hamiltonian chaos and fractional dynamics, NITs RKhD. Izhevskii institut komp'iuternykh issledovanii, M.–Izhevsk, 2010, 472 pp. (in Russian)

[7] Kuznetsov A. P., Savin A. V., Sedova Yu. V., “Bogdanov–Takens bifurcation: from flows to discrete systems”, Izvestiya VUZ. Applied Nonlinear Dynamics, 17:6 (2009), 139–158 (in Russian)

[8] Morozov A. D., Resonances, cycles and chaos in quasiconservative systems, NITs RKhD. Izhevskii institut komp'iuternykh issledovanii, M.–Izhevsk, 2005, 424 pp. (in Russian)

[9] Kapranov M. V., Kuleschov V. N., Utkin G. M., The theory of oscillations in radio engineering, Nauka, M., 1984, 320 pp. (in Russian)

[10] Zaitsev V. V., Stulov I. V., “About influence of the changed harmonics on dynamics of self-oscillations in discrete time”, Izvestiya VUZ. Applied Nonlinear Dynamics, 23:6 (2015), 40–44 (in Russian)

[11] Mishchenko E. F., Sadovnichii V. A., Kolesov A. Yu., Rozov N. Kh., Many-sided chaos, Fizmatlit, M., 2013, 432 pp. (in Russian)

[12] Malakhov A. N., Fluctuations in self-oscillations systems, Nauka, M., 1968, 660 pp. (in Russian)

[13] Zaitsev V. V., Karlov Ar. V., “Discrete mapping of the oscillator with nonlinear disspation and frequency detecting”, Radioengineering, 2014, no. 4, 50–54 (in Russian)

[14] Kornienko V. N., Privezentsev A. P., “Features of the multimode self-consistent dynamics of an ensemble of self-oscillators and the field in a rectangular domain”, Journal of Communications Technology and Electronics, 58:7 (2013), 691–698 (in Russian) | DOI

[15] Jalnine A. Yu., “A new information transfer scheme based on phase modulation of a carrier chaotic signal”, Izvestiya VUZ. Applied Nonlinear Dynamics, 22:5 (2014), 3–12 (in Russian)