Geodesic
The discrete van der Pol oscillator: Finite differences and slow amplitudes
Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 25 (2017) no. 6, pp. 70-78

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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.
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 selfoscillators. 
@article{IVP_2017_25_6_a4,
     author = {V. V. Zaytcev},
     title = {The discrete van der {Pol} oscillator: {Finite} differences and slow amplitudes},
     journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
     pages = {70--78},
     publisher = {mathdoc},
     volume = {25},
     number = {6},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVP_2017_25_6_a4/}
}
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V. V. Zaytcev. The discrete van der Pol oscillator: Finite differences and slow amplitudes. Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 25 (2017) no. 6, pp. 70-78. http://geodesic.mathdoc.fr/item/IVP_2017_25_6_a4/