Bulletin KRASEC. Phys. & Math. Sci. 2016. V. 13. no. 2. pp. 39-45. ISSN 2313-0156

Back to contents

DOI: 10.18454/2313-0156-2016-13-2-39-45

MATHEMATICAL MODELLING

MSC 34C26

MATHEMATICAL MODELING OF NONLINEAR HEREDITARY OSCILLATORS ON THE EXAMPLE OF DUFFING OSCILLATOR WITH FRACTIONAL DERIVATIVES IN THE SENSE OF RIEMANN-LIOUVILLE

I.V. Drobysheva

Vitus Bering Kamchatka State University, 683031, Petropavlovsk-Kamchatsky, Pogranichnaya st., 4, Russia
E-mail: irisha_dr@mail.ru

The paper presents a mathematical hereditary model of Duffing oscillator with friction, which is a generalization of the previously known classical model of Duffing oscillator. This generalization is the replacement of integer derivative by fractional order derivatives in the model equation in the sense of Riemann-Liouville. An explicit finite difference scheme was built to calculate the approximate solution, as well as phase trajectories for different values of control parameters.

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

References

  1. Uchaykin V.V. Metod drobnykh proizvodnykh [Method of fractional derivatives] Ul’yanovsk. Artishok. 2008. 512 p.
    2. 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.
    3. 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.
    4. Petras I. Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. New York. Springer. 2011. 218 p.
    5. 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.
    6. Nakhushev A. M. Drobnoe ischislenie i ego prilozheniya [Fractional calculus and its applications]. Moscow. Fizmatlit. 2003. 272 p.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. Marchuk G.I. Vychislitel’nye metody [Computational Methods]. Moscow. Nauka. 1977. 456 p.
    13. Parovik R. I. Ob issledovanii ustoychivosti ereditarnogo ostsillyatora Van-der-Polya [Investigation of the stability of Van-der-Pol hereditary oscillator]. Fundamental’nye issledovaniya – Fundamental Research. 2016. no. 3-2. pp. 283–287.

For citation: Drobysheva I.V. Mathematical modeling of nonlinear hereditary oscillators on the example of Duffing oscillator with fractional derivatives in the sense of Riemann Liouville. Bulletin KRASEC. Physical and Mathematical Sciences 2016, vol. 13, no 2, 39-45. DOI: 10.18454/2313-0156-2016-13-2-39-45.

Original article submitted: 18.03.2016

Drob

  Drobysheva Irina Viktorovna – graduate student of the second year of training, the master’s program in Applied Mathematics and Computer Science, Vitus Bering Kamchatka State University, Petropavlovsk-Kamchatsky, Russia.

Download article Drobysheva I.V.