Bulletin KRASEC. Phys. & Math. Sci, 2015, V. 10, №. 1, pp. 16-21. ISSN 2313-0156

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DOI: 10.18454/2313-0156-2015-10-1-16-21

MSC 37C70

MATHEMATICAL MODELING OF NONLOCAL OSCILLATORY DUFFING SYSTEM WITH FRACTAL FRICTION

R.I. Parovik¹²

¹Institute of Cosmophysical Researches and RadioWave Propagation Far-Eastern Branch, Russian Academy of Sciences, 684034, Kamchatskiy Kray, Paratunka, Mirnaya st., 7, Russia.
²Vitus Bering Kamchatka State University, 683031, Petropavlovsk-Kamchatsky, Pogranichnaya st., 4, Russia.

E-mail: romanparovik@gmail.com.

The paper considers a nonlinear fractal oscillatory Duffing system with friction. The numerical analysis of this system by a finite-difference scheme was carried out. Phase portraits and system solutions were constructed depending on fractional parameters.

Key words: Gerasimov-Caputo operator, phase portrait, Duffing oscillator, finitedifference scheme.

References

  1. Rekhviashvili S.Sh. Razmernye yavleniya v fizike kondensirovannogo sostoyaniya i nanotekhnologiyakh [Dimentional phenomena in condensed matter physics and nanotechnologies]. Nalchik, KBNTs RAN, 2014. 250 p.
    2. Petras I. Fractional-Order Nonlinear Systems. Modeling, Analysis and Simulation. Berlin-Heidelberg. Springer. 2011. 218 p.
    3. Kao B.G. A Three-Dimensional Dynamic Tire Model for Vehicle Dynamic Simulations. Tire Science and Technology. 2000, vol. 28, no. 2, pp. 72-95. DOI: http://dx.doi.org/10.2346/1.2135995.
    4. Rossikhin Y.A., Shitikova M.V. Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. App. Mech. Rev. 2010. Vol. 63. no. 1. 010801. DOI: http://dx.doi.org /10.1115/1.4000563.
    5. Syta A., Litak G., Lenci S., Scheffler M. Chaotic vibrations of the duffing system with fractional damping. Chaos. 2014. vol. 24. No. 1. 013107. doi: http://dx.doi.org /10.1063/1.4861942.
    6. Sheu L.J., Chen H.K., Tam L.M. Chaotic dynamics of the fractionally damped Duffing equation. Chaos. Solitions. Fractals. 2007. vol. 32, pp. 1459-1468.
    7. Nakhushev A.M. Drobnoe ischislenie i ego primenenie [Fractional calculus and its application]. Moscow, Fizmatlit, 2003. 272 p.
    8. Parovik R.I. Numerical analysis some oscillation equations with fractional order derivatives. Bulletin KRAESC. Phys. & Math. Sci., 2014, vol. 9, no. 2, pp. 34-38.
    9. Ebeling V. Obrazovanie struktur pri neobratimyih protsessah. Vvedenie v teoriyu dissipativnyih struktur [Education structures in irreversible processes. Introduction to the theory of dissipative structures]. Moscow, Mir, 1979. 279 pp.
    10. Grinchenko V.T., Snarskii A., Matsypura V.T. Vvedenie v nelineynuyu dinamiku: Haos i fraktalyi [Introduction to nonlinear dynamics: Chaos and fractals]. Moscow LKI, 2007. 264 pp.

Original article submitted: 13.04.2015


Par

   Parovik Roman Ivanovich – Ph.D. (Phys. & Math.), Dean of the Faculty of Physics and Mathematics Vitus Bering Kamchatka State University, Senior Researcher of Lab. Modeling of Physical Processes, Institute of Cosmophysical Researches and Radio Wave Propagation FEB RAS.

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