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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu


Chaos Synchronization of the Fractional Rucklidge System based on New Adomian Polynomials

Journal of Applied Nonlinear Dynamics 6(3) (2017) 379--385 | DOI:10.5890/JAND.2017.09.006

Guo-Cheng Wu$^{1}$ , Dumitru Baleanu$^{2}$,$^{3}$, Lan-Lan Huang$^{4}$

$^{1}$ College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, China

$^{2}$ Department of Mathematics, Cankaya University, 06530 Balgat, Ankara, Turkey

$^{3}$ Institute of Space Sciences, Magurele–Bucharest, Romania

$^{4}$ College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China

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Abstract

The fractional Rucklidge system is a new kind of chaotic models which hold the feature of memory effects and can depict the long history interactions. A numerical formula is proposed by use of the fast Adomian polynomials. Chaotic behavior are discussed and the Poincare sections are given for various fractional cases. It’s also applied in chaos synchronization of the fractional system.

References

  1. [1]  Rucklidge, A.M. (1992), Chaos in models of double convection, J. Fluid Mech., 237, 209–229.
  2. [2]  Dias, F.S. and Mello, L.F. (2013), Hopf Bifurcations and small amplitude limit cycles in Rucklidge systems, Electron. J. Differ. Equ., 48, 1–9.
  3. [3]  Kocamaz, U.E. and Uyaroğlu, Y. (2014), Controlling Rucklidge chaotic system with a single controller using linear feedback and passive control methods, Nonlinear Dyn., 75, 63–72.
  4. [4]  Zhang, Y. and Zhou, T. (2007), Three schemes to synchronize chaotic fractional-order Rucklidge systems, Int. J. Modern Phys. B., 21, 2033–2044.
  5. [5]  Caputo, M. (1967), Linear model of dissipation whose Q is almost frequency independent–II, Geophys. J. R. Astron. Soc. 13, 529–539.
  6. [6]  Paola, M.D., Pinnola, F.P., Francesco, and Zingales, M. (2013), A discrete mechanical model of fractional hereditary materials, Meccanica, 48, 1573–1586.
  7. [7]  Mainardi, F. and Pagnini, G. (2003), The Wright functions as solutions of the time-fractional diffusion equation, Appl. Math. Comput., 14, 51–62.
  8. [8]  Ji, J. (2015), Discrete fractional diffusion equation with a source term, J. Comput. Complex. Appl., 1, 10–14.
  9. [9]  Pu, Y.F., Zhou, J.L., and Yuan, J.L. (2010), Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement, IEEE Trans. Image Proc., 19, 491–511.
  10. [10]  Li, C.P. and Peng, G.J. (2004), Chaos in Chen’s system with a fractional order, Chaos, Soliton. Fract. 22, 443–450.
  11. [11]  Daftardar–Gejji, V., Bhalekar, S., and Gade, P. (2012), Dynamics of fractional-ordered Chen system with delay, Pramma-J. Phys., 79, 61–69.
  12. [12]  Wu, Z.B. and Zou, Y.Z. (2014), Global fractional–order projective dynamical systems, Commun. Nonlinear Sci. Numer. Simulat., 19, 2811–19
  13. [13]  Allahviranloo, T. and Jamshidi, L. (2009), Solution of fuzzy differential equations under generalized differentiability by Adomian decomposition method, Iran. J. Optim., 1, 57-75.
  14. [14]  Caponetto, R. and Fazzino, S. (2013), An application of Adomian decomposition for analysis of fractional– order chaotic systems, Int. J. Bifur. Chaos, 23, 1350050.
  15. [15]  Daftardar–Gejji, V. and Jafari, H. (2005), Adomian decomposition: a tool for solving a system of fractional differential equations, J. Math. Anal. Appl., 301, 508-518.
  16. [16]  Dehghan, M. and Hashemi, B. (2006), Solution of the fully fuzzy linear systems using the decomposition procedure,Appl. Math. Comput., 182, 1568–1580.
  17. [17]  Evans, D. J. and Raslan, K. (2005), The Adomian decomposition method for solving delay differential equation, Int. J. Comput. Math., 82, 49–54.
  18. [18]  Evirgen, F. and Ozdemir, N. (2011), Multistage Adomian decomposition method for solving NLP problems over a nonlinear fractional dynamical system, J. Comput. Nonlinear Dyn., 6, 021003.
  19. [19]  Duan, J.S. (2010), Recurrence triangle for Adomian polynomials, Appl. Math. Comput., 216, 1235–1241.
  20. [20]  Duan, J.S. (2010), An efficient algorithm for the multivariable Adomian polynomials, Appl. Math. Comput., 217, 2456–2467.
  21. [21]  Duan, J.S. (2011), Convenient analytic recurrence algorithms for the Adomian polynomials, Appl. Math. Comput., 217, 6337–6348.
  22. [22]  Duan, J.S. and Rach, R. (2011), New higher–order numerical one-step methods based on the Adomian and the modified decomposition methods, Appl. Math. Comput., 218, 2810–2828.
  23. [23]  Liu, C.X. and Huang, L.L. (2015), Adomian decomposition method for detection of Chaos in the Rucklidge system, U. Politeh Buch. Ser. A 77, 299–306.
  24. [24]  Adomian, G. (1994), Solving frontier problems of physics the decomposition method, Springer, Germany.
  25. [25]  Zhou, T. and Li, C. (2005), Synchronization in fractional-order differential systems, Physica D, 212, 111–125.
  26. [26]  Deng, W., et al. (2007), Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dyn., 48, 409–416.
  27. [27]  Wang, X.Y. and Song, J.M. (2009), Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control, Commun. Nonlinear Sci. Numer. Simulat., 14, 3351–3357.
  28. [28]  Li, Y., et al. (2010), Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag??Leffler stability, Comput. Math. Appl., 59, 1810–1821.
  29. [29]  Baleanu, D., et al. (2011),Fractional dynamics and control, Springer Science & Business Media.
  30. [30]  Aguila–Camacho, N., et al. (2014), Lyapunov functions for fractional order systems, Commun. Nonlinear Sci. Numer. Simulat., 19, 2951–2957.