Bulletin KRASEC. Phys. & Math. Sci. 2016. V. 13. no. 2. pp. 46-49. ISSN 2313-0156

Back to contents

DOI: 10.18454/2313-0156-2016-13-2-46-49

MSC 34C26

DUFFING OSCILLATOR WITH EXTERNAL HARMONIC ACTION AND VARIABLE FRACTIONAL RIEMANN-LIOUVILLE DERIVATIVE CHARACTERIZING VISCOUS FRICTION

V. A. Kim

Vitus Bering Kamchatka State University, 683031, Petropavlovsk-Kamchatsky, Pogranichnaya st., 4, Russia

E-mail: valentinekim93@mail.ru

The paper suggested generalization of Duffing oscillator with viscous hereditary friction which is represented by the operator of a variable fractional derivative in the sense of Riemann-Liouville. Explicit finite-difference scheme was derived to calculate approximate solutions, and phase trajectories for different values of control parameters were plotted.

Key words: Riemann-Liouville derivative, Grunwald-Letnikov derivative, heredity, Duffing oscillator, phase trajectory.

References

  1. Petras I. Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. New York. Springer. 2011. 218 p.
    2. Uchaykin V.V. Metod drobnykh proizvodnykh [Fractional derivative method]. Ul’yanovsk. Artishok. 2008. 512 p.
    3. Syta A., Litak G., Lenci S., Scheffler M. Chaotic vibrations of the Duffing system with fractional damping. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2014. vol. 24. no. 1. 013107
    4. Gao X., Yu J. Chaos in the fractional order periodically forced complex Duffing’s oscillators. Chaos, Solitons & Fractals. 2005. vol. 24. no. 4. pp. 1097–1104.
    5. Rossikhin Y. A., Shitikova M. V. Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Applied Mechanics Reviews. 2010. vol. 63. no. 1. 010801
    6. Parovik R. I. Mathematical modeling of nonlocal oscillatory Duffing system with friction. Bulletin KRASEC. Phys. & Math. Sci. 2015. vol. 10. no. 1. pp. 16–21.
    7. Parovik R. I. O chislennom reshenii uravneniya fraktal’nogo ostsillyatora s proizvodnoy drobnogo peremennogo poryadka ot vremeni [Numerical solution of fractal oscillatory equation with time derivative of fractional variable order]. Vestnik KRAUNTs. Fiz.-mat. nauki – Bulletin KRASEC. Phys. & Math. Sci. 2014. no. 1(8). pp. 60–65.
    8. Parovik R. I. Numerical analysis of some oscillatory equations with fractional derivative. Bulletin KRASEC. Phys. & Math. Sci. 2014. vol. 9. no. 2. pp. 34–38.
    9. Parovik R. I. Ob odnoy konechno-raznostnoy skheme dlya matematicheskoy modeli nelineynogo ereditarnogo ostsillyatora [A finite-difference scheme for a mathematical model of nonlinear hereditary oscillator]. Mezhdunarodnyy nauchno-issledovatel’skiy zhurnal – International Research Journal. 2016. no. 4-2(9). pp. 138–142.
    10. Petukhov A. A., Reviznikov D. L. Algoritmy chislennykh resheniy drobno-differentsial’nykh uravneniy [Algorithms of numerical solutions of fractional-differential equations]. Vestnik Moskovskogo aviatsionnogo instituta – Bulletin of the Moscow Aviation Institute. 2009. vol. 16. no. 6. pp. 228–243.
    11. Marchuk G.I. Vychislitel’nye metody [Calculation methods]. Moscow. Nauka. 1977. 456 p.
    12. Parovik R. I. Ob issledovanii ustoychivosti ereditarnogo ostsillyatora Van-der-Polya [On the investigation of hereditary Van-der-Pol oscillator]. Fundamental’nye issledovaniya – Fundamental Research. 2016. no. 3-2. pp. 283–287.

For citation: Kim V. A. Duffing oscillator with external harmonic action and variable fractional Riemann-Liouville derivative characterizing viscous friction. Bulletin KRASEC. Physical and Mathematical Sciences 2016, vol. 13, no 2, 46-49. DOI: 10.18454/2313-0156-2016-13-2-46-49.

Original article submitted: 16.04.2016

   

Kim

   Kim Valentin Aleksandrovich – fourthyear student training direction «Applied Mathematics and Informatics», Kamchatka Vitus Bering State University, Petropavlovsk- Kamchatsky, Russia.

Download article Kim V.A.