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Dynamics of distributed-order hyperchaotic complex van der Pol oscillators and their synchronization and control

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The distributed-order hyperchaotic unforced and forced complex van der Pol oscillators with complex parameter are introduced and investigated in this paper. The basic dynamical properties including equilibrium point and its stability and chaotic behavior of the unforced oscillator are studied. The intervals of the parameters values at which this oscillator has periodic, chaotic, and hyperchaotic behaviors are calculated using Lyapunov exponents. These intervals of chaotic and hyperchaotic behaviors can be used in many applications such as secure communication and electronic circuits. Using the linear feedback control, the control of solutions of our oscillator(unforced) converge to a fixed point are studied. We state a scheme to achieve the complete synchronization between two distributed-order hyperchaotic unforced complex van der Pol oscillators. The analytical formula of the controller is derived and used to achieve synchronization. Secure communications via hyperchaotic masking for a text which contains alphabets, numbers, space, and symbols are investigated using the proposed scheme of this work. The dynamics of the distributed-order hyperchaotic forced complex van der Pol oscillator with complex parameter is investigated. Synchronization and secure communications can be similarly studied for the forced oscillator.

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  1. 1.

    M. Caputo, Elasticita e dissipazione, Zanichelli (1969)

  2. 2.

    Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59, 1810–1821 (2010)

  3. 3.

    Y. Li, Y. Chen, I. Podlubny, Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965–1969 (2009)

  4. 4.

    N. Aguila-Camacho, M.A. Duarte-Mermoud, J.A. Gallegos, Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19, 2951–2957 (2014)

  5. 5.

    G. Fernandez-Anaya, G. Nava-Antonio, J. Jamous-Galante, R. Muñoz-Vega, E. Hernández-Martínez, Lyapunov functions for a class of nonlinear systems using Caputo derivative. Commun. Nonlinear Sci. Numer. Simul. 43, 91–99 (2017)

  6. 6.

    M. Caputo, Mean fractional-order-derivatives differential equations and filters. Ann. Univ. Ferrara 41, 73–84 (1995)

  7. 7.

    M. Caputo, Distributed order differential equations modelling dielectric induction and diffusion. Fraction. Calc. Appl. Anal. 4, 421–442 (2001)

  8. 8.

    R. Bagley, P. Torvik, On the existence of the order domain and the solution of distributed order equations-part I. Int. J. Appl. Math. 2, 865–882 (2000)

  9. 9.

    R. Bagley, P. Torvik, On the existence of the order domain and the solution of distributed order equations-part II. Int. J. Appl. Math. 2, 965–988 (2000)

  10. 10.

    G. Fernández-Anaya, G. Nava-Antonio, J. Jamous-Galante, R. Muñoz-Vega, E. Hernández-Martínez, Asymptotic stability of distributed order nonlinear dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 48, 541–549 (2017)

  11. 11.

    G.M. Mahmoud, T. Aboelenen, T.M. Abed-Elhameed, A.A. Farghaly, Generalized Wright stability for distributed fractional-order nonlinear dynamical systems and their synchronization. Nonlinear Dyn. 97, 413–429 (2019)

  12. 12.

    T. Atanackovic, M. Budincevic, S. Pilipovic, On a fractional distributed-order oscillator. J. Phys. A Math. Gen. 38, 6703 (2005)

  13. 13.

    M.S. Tavazoei, Fractional/distributed-order systems and irrational transfer functions with monotonic step responses. J. Vib. Control 20, 1697–1706 (2014)

  14. 14.

    T. Aboelenen, Local discontinuous Galerkin method for distributed-order time and space-fractional convection–diffusion and Schrödinger-type equations. Nonlinear Dyn. 92, 395–413 (2018)

  15. 15.

    J. Chen, C. Li, X. Yang, Chaos synchronization of the distributed-order Lorenz system via active control and applications in chaotic masking. Int. J. Bifurc. Chaos 28, 1850121 (2018)

  16. 16.

    L.M. Pecora, T.L. Carroll, Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)

  17. 17.

    G.M. Mahmoud, E.E. Mahmoud, A.A. Arafa, On projective synchronization of hyperchaotic complex nonlinear systems based on passive theory for secure communications. Phys. Scr. 87, 055002 (2013)

  18. 18.

    G.M. Mahmoud, E.E. Mahmoud, A.A. Arafa, Projective synchronization for coupled partially linear complex-variable systems with known parameters. Math. Methods Appl. Sci. 40, 1214–1222 (2017)

  19. 19.

    G.M. Mahmoud, A.A. Farghaly, T.M. Abed-Elhameed, M.M. Darwish, Adaptive dual synchronization of chaotic (hyperchaotic) complex systems with uncertain parameters and its application in image encryption. Acta Phys. Pol. B 49, 1923 (2018)

  20. 20.

    T. Carletti, R. Serra, I. Poli, M. Villani, A. Filisetti, Sufficient conditions for emergent synchronization in protocell models. J. Theor. Biol. 254, 741–751 (2008)

  21. 21.

    G.M. Mahmoud, T.M. Abed-Elhameed, M.E. Ahmed, Generalization of combination-combination synchronization of chaotic n-dimensional fractional-order dynamical systems. Nonlinear Dyn. 83, 1885–1893 (2016)

  22. 22.

    G.M. Mahmoud, M.E. Ahmed, T.M. Abed-Elhameed, Active control technique of fractional-order chaotic complex systems. Eur. Phys. J. Plus 131, 200 (2016)

  23. 23.

    S. Bowong, F.M. Kakmeni, Synchronization of uncertain chaotic systems via backstepping approach. Chaos Solitons Fractals 21, 999–1011 (2004)

  24. 24.

    G.M. Mahmoud, M.E. Ahmed, T.M. Abed-Elhameed, On fractional-order hyperchaotic complex systems and their generalized function projective combination synchronization. Opt. Int. J. Light Electron Opt. 130, 398–406 (2017)

  25. 25.

    Z. Gao, Y. Wang, L. Zhang, Y. Huang, W. Wang, The dynamic behaviors of nodes driving the structural balance for complex dynamical networks via adaptive decentralized control. Int. J. Mod. Phys. B 32, 1850267 (2018)

  26. 26.

    E.A. Jackson, I. Grosu, An open-plus-closed-loop (OPCL) control of complex dynamic systems. Phys. D 85, 1–9 (1995)

  27. 27.

    M. Rafikov, J.M. Balthazar, On control and synchronization in chaotic and hyperchaotic systems via linear feedback control. Commun. Nonlinear Sci. Numer. Simul. 13, 1246–1255 (2008)

  28. 28.

    C. Edwards, S. Spurgeon, Sliding Mode Control: Theory and Applications (CRC Press, Boca Raton, 1998)

  29. 29.

    S. Liu, Q. Wang, Outer synchronization of general colored networks with different-dimensional node via sliding mode control. Int. J. Mod. Phys. B 32, 1850342 (2018)

  30. 30.

    G.M. Mahmoud, A.A. Arafa, T.M. Abed-Elhameed, E.E. Mahmoud, Chaos control of integer and fractional orders of chaotic Burke–Shaw system using time delayed feedback control. Chaos Solitons Fractals 104, 680–692 (2017)

  31. 31.

    G.M. Mahmoud, T.M. Abed-Elhameed, A.A. Farghaly, Double compound combination synchronization among eight n-dimensional chaotic systems. Chin. Phys. B 27, 080502 (2018)

  32. 32.

    B. Van der Pol, J. Van Der Mark, Frequency demultiplication. Nature 120, 363 (1927)

  33. 33.

    G.M. Mahmoud, A.A. Farghaly, Chaos control of chaotic limit cycles of real and complex van der Pol oscillators. Chaos Solitons Fractals 21, 915–924 (2004)

  34. 34.

    K. Chung, C. Chan, Z. Xu, G. Mahmoud, A perturbation-incremental method for strongly nonlinear autonomous oscillators with many degrees of freedom. Nonlinear Dyn. 28, 243–259 (2002)

  35. 35.

    Y. Xu, W. Xu, G.M. Mahmoud, On a complex beam–beam interaction model with random forcing. Phys. A 336, 347–360 (2004)

  36. 36.

    G.M. Mahmoud, E.E. Mahmoud, M.E. Ahmed, A hyperchaotic complex Chen system and its dynamics. Int. J. Appl. Math. Stat. 12, 90–100 (2007)

  37. 37.

    C. Huang, Multiple scales scheme for bifurcation in a delayed extended van der Pol oscillator. Phys. A 490, 643–652 (2018)

  38. 38.

    M. Yorinaga et al., Bifurcation of a periodic solution of van der Pol’s equation with the harmonic forcing term. J. Sci. Hiroshima Univ. Ser. AI (Mathematics) 26, 51–70 (1962)

  39. 39.

    H.Y. Hafeez, C.E. Ndikilar, S. Isyaku, Analytical study of the van der Pol equation in the autonomous regime. Progress 11, 252–262 (2015)

  40. 40.

    J. Guckenheimer, K. Hoffman, W. Weckesser, The forced van der Pol equation I: The slow flow and its bifurcations. SIAM J. Appl. Dyn. Syst. 2, 1–35 (2003)

  41. 41.

    M.A. Barron, Stability of a ring of coupled van der Pol oscillators with non-uniform distribution of the coupling parameter. J. Appl. Res. Technol. 14, 62–66 (2016)

  42. 42.

    C.M. Pinto, J.T. Machado, Complex order van der Pol oscillator. Nonlinear Dyn. 65, 247–254 (2011)

  43. 43.

    R.S. Barbosa, J.T. Machado, B. Vinagre, A. Calderon, Analysis of the Van der Pol oscillator containing derivatives of fractional order. J. Vib. Control 13, 1291–1301 (2007)

  44. 44.

    A. Kimiaeifar, A. Saidi, G. Bagheri, M. Rahimpour, D. Domairry, Analytical solution for Van der Pol-Duffing oscillators. Chaos Solitons Fractals 42, 2660–2666 (2009)

  45. 45.

    S. Wen, Y. Shen, X. Li, S. Yang, Dynamical analysis of Mathieu equation with two kinds of van der Pol fractional-order terms. Int. J. Non-Linear Mech. 84, 130–138 (2016)

  46. 46.

    S.R. Munjam, R. Seshadri, Analytical solutions of nonlinear system of fractional-order Van der Pol equations. Nonlinear Dyn. 95, 2837–2854 (2019)

  47. 47.

    K.S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations (1993)

  48. 48.

    D.G. Duffy, Transform Methods for Solving Partial Differential Equations (CRC, Boca Raton, 2004)

  49. 49.

    A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov exponents from a time series. Phys. D 16, 285–317 (1985)

  50. 50.

    G.M. Mahmoud, M. Al-Kashif, S.A. Aly, Basic properties and chaotic synchronization of complex Lorenz system. Int. J. Mod. Phys. C 18, 253–265 (2007)

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The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Research Groups Program under grant number R. G. P. 2/19/40.

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Correspondence to Gamal M. Mahmoud.

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Mahmoud, G.M., Farghaly, A.A., Abed-Elhameed, T.M. et al. Dynamics of distributed-order hyperchaotic complex van der Pol oscillators and their synchronization and control. Eur. Phys. J. Plus 135, 32 (2020). https://doi.org/10.1140/epjp/s13360-019-00006-1

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