For citation:
Zaitsev V. V. The discrete van der paul oscillator: finite differences and slow amplitudes. Izvestiya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, iss. 6, pp. 70-78. DOI: 10.18500/0869-6632-2017-25-6-70-78
The discrete van der paul oscillator: finite differences and slow amplitudes
For sampling of time in a differential equation of movement of van der Pol oscillator (generator) it is offered to use a combination of the numerical method of finite differences and the asymptotic method of the slowl-changing amplitudes. The difference approximations of temporal derivatives are selected so that, first, to save conservatism and natural frequency of the linear circuit of self-oscillatory system in the discrete time. Secondly, coincidence of the difference shortened equation for the complex amplitude of self-oscillations in the discrete time with Euler’s approximation of the shortened equation for amplitude of self-oscillations in analog system prototype is required. It is shown that realization of such approach allows to create discrete mapping of the van der Pol oscillator and a number of mappings of Thomson type oscillators. The adequacy of discrete models to analog prototypes is confirmed with also numerical experiment.
- Kuznetsov A.P., Kuznetsov S.P., Ryskin N.M. Nelinejnye Kolebaniya. M.: Fizmatlit, 2005. 292 p. (in Russian).
- Kuznetsov A.P., Seliverstova E.S., Trubetskov D.I., Turukina L.V. Phenomenon of the van der Pol equation. Izvestiya VUZ. Applied Nonlinear Dynamics. 2014. Vol. 22, Issue 4. Pp. 3–42 (in Russian).
- Oppenheim A., Schafer R. Discrete-Time Signal Processing. Prentice-Hall Inc., New Jersey, 1999. 870 p.
- Zaslavsky G.M. Hamiltonian Chaos and Fractional Dynamics. Oxford: University Press, 2005.
- Kuznetsov A.P., Turukina L.V. Synchronization of self-oscillating van der Pol– Dyuffing system by the short pulses. Izvestiya VUZ. Applied Nonlinear Dynamics. 2004. Vol. 12, Issue 5. Pp. 16–31 (in Russian).
- Zaitsev V.V., Davydenko S.V, Zaitsev O.V. Dynamics of self-oscillations of the discrete van der Pol oscillator. Fizika Volnovykh Protsessov i Radiotekhnicheskie Sistemy. 2000. Vol. 3, Issue 2. Pp. 64–67 (in Russian).
- Kuznetsov A.P., Savin A.V., Sedova Yu.V. Bogdanov–Takens bifurcation: From flows to discrete systems. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009. Vol. 17, Issue 6. Pp. 139–158 (in Russian).
- Morozov A.D. Resonances, Cycles and Chaos in Quasi-conservative Systems. Moscow; Izhevsk: Regular and Chaotic Dynamics, 2005. 424 p. (in Russian).
- Zaitsev V.V., Fedyunin E.Yu., Shilin A.N. Finite differences for design of nonlinear discrete time oscillators. Fizika Volnovykh Protsessov i Radiotekhnicheskie Sistemy. 2017. Vol. 20, Issue 2. Pp. 35–41 (in Russian).
- Kapranov M.V., Kuleshov V.N., Utkin G.M. The Theory of Oscillations in Radio Engineering. M.: Nauka, 1984. 320 p. (in Russian).
- Zaitsev V.V. About discrete mapping the van der Pol oscillator. Fizika Volnovykh Protsessov i Radiotekhnicheskie Sistemy. 2014. Vol. 17, Issue 1. Pp. 35–40 (in Russian).
- Lindsey W. Synchronization Systems in Communication and Control. New Jersy: Prentice Hall, 1972. 695 p.
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