Nonlinear Dynamics

, Volume 88, Issue 4, pp 3023–3041 | Cite as

Fractional modified Duffing–Rayleigh system and its synchronization

Original Paper
  • 199 Downloads

Abstract

Chaotic vibrations, stability and synchronization are important topics in nonlinear dynamics, and thus are studied in this paper for a new chaotic system with quadratic and cubic nonlinearly. The modified Duffing–Rayleigh system with piecewise quadratic function is presented firstly. Then, by taking the Melnikov function method, necessary conditions for chaotic motion of the modified Duffing–Rayleigh system are given. Fractional modified Duffing–Rayleigh oscillator is also discussed, and results of computer simulation demonstrate the chaotic dynamic behaviors of the fractional-order modified Duffing–Rayleigh system with order less than 2. Furthermore, generalized projective synchronization of two fractional modified Duffing–Rayleigh oscillators is explored by the active control technology. Adaptive synchronization and parameter identification of a fractional modified Duffing–Rayleigh oscillator with unknown parameters are also investigated. Numerical results validate the effectiveness and applicability of the proposed synchronization schemes.

Keywords

Modified Duffing–Rayleigh oscillator Melnikov function method Fractional-order Synchronization 

Notes

Acknowledgements

This work was supported by Grants from the National Natural Science Foundation of China (Nos. 11526109, 61379021), Natural Science Foundation of Fujian (Nos. 2015J05011, 2016J01671, JK2014028, JA14200), and the outstanding youth foundation of the Education Department of Fujiang Province.

References

  1. 1.
    Agrawal, S.K., Das, S.: A modified adaptive control method for synchronization of some fractional chaotic systems with unknown parameters. Nonlinear Dyn. 73, 907–919 (2013)CrossRefMATHGoogle Scholar
  2. 2.
    Ahmed, E., EI-Sayed, A.M., EI-Saka, H.: Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models. J. Math. Anal. Appl. 325(1), 542–553 (2007)CrossRefMATHGoogle Scholar
  3. 3.
    Bhalekar, S., Daftardar-Gejji, V.: Synchronization of different fractional order chaotic systems using active control. Commun. Nonlinear Sci. Numer. Simul. 15, 3536–3546 (2010)CrossRefMATHGoogle Scholar
  4. 4.
    Brzeziński, D.W.: Accuracy problems of numerical calculation of fractional order derivatives and integrals applying the Riemann–Liouville/Caputo formulas. Appl. Math. Nonlinear Sci. 1(1), 23–44 (2016)CrossRefGoogle Scholar
  5. 5.
    Butzer, P.L., Westphal, U.: An Introduction to Fractional Calculus. World Scientific, Singapore (2000)CrossRefMATHGoogle Scholar
  6. 6.
    Deng, W., Li, C.P.: Chaos synchronization of the fractional Lü system. Phys. A 353, 61–72 (2005)CrossRefGoogle Scholar
  7. 7.
    Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)CrossRefMATHGoogle Scholar
  8. 8.
    Diethelm, K., Freed, A.D., Ford, N.J.: A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1–4), 3–22 (2002)CrossRefMATHGoogle Scholar
  9. 9.
    Friedrich, C.: Relaxation and retardation functions of the Maxwell model with fractional derivatives. Rheol. Acta 30(2), 151–158 (1991)CrossRefGoogle Scholar
  10. 10.
    Ge, Z.M., Zhang, A.R.: Chaos in a modified van der Pol system and in its fractional order systems. Chaos Solitons Fractals 32(5), 1791–1822 (2007)CrossRefGoogle Scholar
  11. 11.
    Guirao, J.L.G., Llibre, J., Vera, J.A.: Periodic solutions induced by an upright position of small oscillations of a sleeping symmetrical gyrostat. Nonlinear Dyn. 73(1), 417–425 (2013)CrossRefMATHGoogle Scholar
  12. 12.
    Hifer, R.: Applications of Fractional Calculus in Physics. World Scientific, Hackensack (2001)Google Scholar
  13. 13.
    Jia, H.Y., Chen, Z.Q., Xue, W.: Analysis and circuit implementation for the fractional-order Lorenz system. Acta Phys. Sin. 62(14), 140503 (2013). (in Chinese)Google Scholar
  14. 14.
    Jiang, J.F., Cao, D.Q., Chen, H.T.: Boundary value problems for fractional differential equation with causal operators. Appl. Math. Nonlinear Sci. 1(1), 11–22 (2016)CrossRefGoogle Scholar
  15. 15.
    Li, C.G., Chen, G.: Chaos and hyperchaos in the fractional-order Rössler equations. Phys. A 341, 55–61 (2004)CrossRefGoogle Scholar
  16. 16.
    Li, C.P., Peng, G.J.: Chaos in Chen’s system with a fractional order. Chaos Solitons Fractals 22, 443–450 (2004)CrossRefMATHGoogle Scholar
  17. 17.
    Li, C.P., Deng, W.H., Xu, D.: Chaos synchronization of the Chua system with a fractional order. Phys. A 360, 171–185 (2006)CrossRefGoogle Scholar
  18. 18.
    Liu, J., Liu, S.T., Yuan, C.H.: Modified generalized projective synchronization of fractional-order chaotic Lü systems. Adv. Differ. Equ. 2013, 374 (2013)CrossRefMATHGoogle Scholar
  19. 19.
    Liu, X., Hong, L., Yang, L.: Fractional-order complex \(T\) system: bifurcations, chaos control, and synchronization. Nonlinear Dyn. 75(3), 589–602 (2014)CrossRefMATHGoogle Scholar
  20. 20.
    López, M.A., Martínez, R.: A note on the generalized Rayleigh equation: limit cycles and stability. J. Math. Chem. 51(4), 1164–1169 (2013)CrossRefMATHGoogle Scholar
  21. 21.
    Martin, K.-R.: Lyapunov function. From MathWorld—A Wolfram Web Resource, created by E.W. Weisstein. http://mathworld.wolfram.com/LyapunovFunction.html
  22. 22.
    Mahmouda, G.M., Mohameda, A.A., Alyb, S.A.: Strange attractors and chaos control in periodically forced complex Duffing’s oscillators. Phys. A 292, 193–206 (2001)CrossRefGoogle Scholar
  23. 23.
    Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems Applications, IMACS, IEEE-SMC 2, pp. 963–968. Lille (1996)Google Scholar
  24. 24.
    Nonnenmacher, T.F., Glöckle, W.G.: A fractional model for mechanical stress relaxation. Philos. Mag. Lett. 64(2), 89–93 (1991)CrossRefGoogle Scholar
  25. 25.
    Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. In: Mathematics in Science and Engineering, Vol. 198. Academic Press, San Diego (1999)Google Scholar
  26. 26.
    Razminia, A., Baleanu, D.: Complete synchronization of commensurate fractional order chaotic systems using sliding mode control. Mechatronics 23(7), 873–879 (2013)CrossRefGoogle Scholar
  27. 27.
    Robinson, C.: Dynamical Systems: Stability, Sympolic Dynamics, and Chaos. CRC, Boca Raton (1995)Google Scholar
  28. 28.
    Si, G., Sun, Z., Zhang, Y., Chen, W.: Projective synchronization of different fractional-order chaotic systems with non-identical orders. Nonlinear Anal.: Real World Appl. 13(4), 1761–1771 (2012)CrossRefMATHGoogle Scholar
  29. 29.
    Siewe, M.S., Tchawoua, C., Woafo, P.: Melnikov chaos in a periodically driven Rayleigh–Duffing oscillator. Mech. Res. Commun. 37(4), 363–368 (2010)CrossRefMATHGoogle Scholar
  30. 30.
    Song, L., Yang, J.Y., Xu, S.Y.: Chaos synchronization for a class of nonlinear oscillators with fractional order. Nonlinear Anal. 72, 2326–2336 (2010)CrossRefMATHGoogle Scholar
  31. 31.
    Tang, K.-S., Man, K.F., Zhong, G.-Q., Chen, G.R.: Generating chaos via \(x|x|\). IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 48(5), 636–641 (2001)CrossRefMATHGoogle Scholar
  32. 32.
    Ueta, T., Kawakami, H.: Unstable saddle-node connecting orbits in the averaged Duffing–Rayleigh equation. In: Proceeding of IEEE International Symposium on Circuits and Systems, Vol. 3, pp. 288–291 (1996)Google Scholar
  33. 33.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990)CrossRefMATHGoogle Scholar
  34. 34.
    Wu, X.J., Lu, Y.: Generalized projective synchronization of the fractional-order Chen hyperchaotic system. Nonlinear Dyn. 57, 25–35 (2009)CrossRefMATHGoogle Scholar
  35. 35.
    Yan, J.P., Li, C.P.: Generalized projective synchronization of a unified chaotic system. Chaos Solitons Fractals 26, 1119–1124 (2005)CrossRefMATHGoogle Scholar
  36. 36.
    Zeng, C.B., Yang, Q.G., Wang, J.W.: Chaos and mixed synchronization of a new fractional-order system with one saddle and two stable node-foci. Nonlinear Dyn. 65, 457–466 (2011)CrossRefMATHGoogle Scholar
  37. 37.
    Zhang, Y.L., Luo, M.K.: Fractional Rayleigh–Duffing-like system and its synchronization. Nonlinear Dyn. 70, 1173–1183 (2012)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.College of ComputerMinnan Normal UniversityZhangzhouChina
  2. 2.School of Mathematics and StatisticsMinnan Normal UniversityZhangzhouChina

Personalised recommendations