Physically reasonable time sampling in mathematical models of generators of regular and chaotic oscillations


Cite item

Abstract

The issue of transition to discrete time in mathematical models of nonlinear dynamic systems oscillating in continuous time is considered. On the basis of the examples of the Dmitriev – Kislov and van der Pol generators, the approach based on maintaining in the process of time sampling the impulse response of the linear oscillatory circuit included in the generator is described. This «physically reasonable» sampling allows models of non-linear dynamics to be formulated in discrete time, adequately reproducing the characteristics of analog prototypes, which is not always possible with a widely used combination of explicit and implicit Euler methods.

About the authors

V.V. Zaitsev

Samara National Research University

Author for correspondence.
Email: zaitsev@samsu.ru

References

  1. Chirikov B.V. Issledovaniya po teorii nelineynogo rezonansa i stohastichnosti [Research on the theory of nonlinear resonance and stochasticity]. Novosibirsk: IYAF SO AN SSSR, 1969. 314 p. [in Russian].Arrowsmith D.K. [et al.] The Bogdanov map: bifurcations, mode locking, and chaos in a dissipative system. International Journal of Bifurcation and Chaos, 1993, vol. 3, no. 4, pp. 803–842 [in English].Kuznetsov A.P., Savin A.V., Sedova Yu.V. Bifurkatsiya Bogdanova – Takensa: ot nepreryvnoy k diskretnoy modeli [Bogdanov – Takens bifurcation:from flows to discrete systems]. Izvestiya vuzov. Prikladnaya nelineynaya dinamika [Izvestiya VUZ. Applied nonlinear dynamics], 2009, vol. 17, no. 6, pp. 39–158 [in Russian].Morozov A.D. Rezonansy, tsikly i haos v kvazikonservativnyh sistemah [Resonances, cycles and chaos in quasi-conservative systems]. M.; Izhevsk: NITS RHD, Izhevskiy institut komp’yuternyh issledovaniy, 2005. 424 p. [in Russian].Shahtarin B.I. [et al.] Generatory haoticheskih kolebaniy [Generators of chaotic oscilations]. M.: Gelios ARV, 2007. 248 p. [in Russian].Zaitsev V.V., Karlov A.V., Fedyunin E.Yu. O diskretnyh modelyah kolebatel’nyh sistem [About discrete models of oscillating systems]. Fizika volnovyh protsessov i radiotehnicheskie sistemy [Physics of wave processes and radio systems], 2015, vol. 18, no. 1, pp. 38–43 [in Russian].Zaitsev V.V. Diskretnyi ostsillyator van der Polya: konechnye raznosti i medlennye amplitudy [The discrete van der Pol oscillator: finite differences and slow amplitudes]. Izvestiya vuzov. Prikladnaya nelinejnaya dinamika [Izvestiya VUZ. Appled nonlinear dynamics], 2017, vol. 25, no. 6, pp. 70–78. DOI: https://doi.org/10.18500/0869-6632-2017-25-6-70-78 [in Russian].Dmitriev A.S., Kislov V.Ya. Stohasticheskie kolebaniya v radiofizike i elektronike [Stochastic oscillations in radiophysics and electro­nics]. M.: Nauka, 1989. 280 p. [in Russian].Oppenheim A., Schafer R. Discrete-time signal processing. Prentice-Hall Inc., New Jersey, 1999. 870 p. [in English].Landa P.S. Nelineynye kolebaniya i volny. Izd. 2-e. [Nonlinear oscilations and waves. 2th edition]. M.: Librokom, 2010. 552 p. [in Russian].Dmitriev A.S., Panas A.I. Dinamicheskiy haos: novye nositeli informatsii dlya sistem svyazi [Dynamic chaos: new media for communication systems]. M.: Fizmatlit, 2002. 252 p. [in Russian].

Copyright (c) 2019 Zaitsev V.

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