THE DISCRETE VAN DER PAUL OSCILLATOR: FINITE DIFFERENCES AND SLOW AMPLITUDES
Cite this article as:
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: https://doi.org/10.18500/0869-6632-2017-25-6-70-78
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.
DOI: 10.18500/0869-6632-2017-25-6-70-78
References: Zaitsev V.V. The discrete van der Paul oscillator: Finite differences and slow amplitudes. Izvestiya VUZ. Applied Nonlinear Dynamics. 2017. Vol. 25. Issue 6. P. 70–78. DOI: 10.18500/0869-6632-2017-25-6-70-78
BibTeX
author = {V. V. Zaitsev},
title = {THE DISCRETE VAN DER PAUL OSCILLATOR: FINITE DIFFERENCES AND SLOW AMPLITUDES},
year = {2017},
journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
volume = {25},number = {6},
url = {https://old-andjournal.sgu.ru/en/articles/the-discrete-van-der-paul-oscillator-finite-differences-and-slow-amplitudes},
address = {Саратов},
language = {russian},
doi = {10.18500/0869-6632-2017-25-6-70-78},pages = {70--78},issn = {0869-6632},
keywords = {self-oscillatory system,van der Pol’s equation,the discrete time,finite differences,slowly changing amplitudes,the shortened equations,the discrete mapping of Thomson self-oscillators.},
abstract = {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. DOI: 10.18500/0869-6632-2017-25-6-70-78 References: Zaitsev V.V. The discrete van der Paul oscillator: Finite differences and slow amplitudes. Izvestiya VUZ. Applied Nonlinear Dynamics. 2017. Vol. 25. Issue 6. P. 70–78. DOI: 10.18500/0869-6632-2017-25-6-70-78 }}