Advertisement

Fractional-order two-component oscillator: stability and network synchronization using a reduced number of control signals

  • Romanic Kengne
  • Robert Tchitnga
  • Alain Kammogne Soup Tewa
  • Grzegorz LitakEmail author
  • Anaclet Fomethe
  • Chunlai Li
Open Access
Regular Article
  • 122 Downloads

Abstract

In this paper, a fractional-order version of a chaotic circuit made simply of two non-idealized components operating at high frequency is presented. The fractional-order version of the Hopf bifurcation is found when the bias voltage source and the fractional-order of the system increase. Using Adams–Bashforth–Moulton predictor–corrector scheme, dynamic behaviors are displayed in two complementary types of stability diagrams, namely the two-parameter Lyapunov exponents and the isospike diagrams. The latest being a more fruitful type of stability diagrams based on counting the number of spikes contained in one period of the periodic oscillations. These two complementary types of stability diagrams are reported for the first time in the fractional-order dynamical systems. Furthermore, a new fractional-order adaptive sliding mode controller using a reduced number of control signals was built for the stabilization of a fractional-order complex dynamical network. Two examples are shown on a fractional-order complex dynamical network where the nodes are made of fractional-order two-component circuits. Firstly, we consider an ideal channel, and secondly, a non ideal one. In each case, increasing of the coupling strength leads to the phase transition in the fractional-order complex network.

Graphical abstract

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999) Google Scholar
  2. 2.
    R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000) Google Scholar
  3. 3.
    H. Sun, A. Abdelwahed, B. Onaral, IEEE Trans. Autom. Control 29, 441 (1984) CrossRefGoogle Scholar
  4. 4.
    K. Diethelm, N.J. Ford, A.D. Freed, Nonlinear Dyn. 29, 3 (2002) CrossRefGoogle Scholar
  5. 5.
    A. Atangana, I. Koca, Chaos Soliton. Fract. 89, 1 (2016) CrossRefGoogle Scholar
  6. 6.
    C.A. Monje, Y. Chen, B.M. Vinagre, D. Xue, V. Feliu-Battle, Fractional-order Systems and Controls: Fundamentals and Applications (Springer, London, 2010) Google Scholar
  7. 7.
    I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulations (Springer, Berlin, Heidelberg, 2011) Google Scholar
  8. 8.
    J.C. Wang, J. Electrochem. Soc. 134, 1915 (1987) CrossRefGoogle Scholar
  9. 9.
    R.L. Bagley, R.A. Calico, J. Guid. Control Dyn. 14, 304 (1991) ADSCrossRefGoogle Scholar
  10. 10.
    B. Ducharne, G. Sebald, D. Guyomar, G. Litak, J. Appl. Phys. 117, 243907 (2015) ADSCrossRefGoogle Scholar
  11. 11.
    W.M. Ahmad, R. El-Khazali, Chaos Soliton. Fract. 33, 1367 (2007) ADSCrossRefGoogle Scholar
  12. 12.
    L. Song, S. Xu, J. Yang, Commun. Nonlinear Sci. Numer. Simul. 15, 616 (2010) ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    E. Ahmed, A.S. Elgazzar, Physica A 379, 607 (2007) ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    B.S.T. Alkahtani, A. Atangana, Entropy 18, 100 (2016) ADSCrossRefGoogle Scholar
  15. 15.
    H.M. Baskonus, Z. Hammouch, H. Bulut, Chaos in the fractional order logistic delay system: Circuit realization and synchronization, AIP Conf. Proc. 1738, 290005 (2016) CrossRefGoogle Scholar
  16. 16.
    H.M. Baskonus, T. Mekkaoui, Z. Hammouch, H. Bulut, Entropy 17, 5771 (2015) ADSCrossRefGoogle Scholar
  17. 17.
    R. Kengne, R. Tchitnga, S. Mabekou, B.R. Wafo Tekam, G.B. Soh, A. Fomethe, Chaos Soliton. Fract. 111, 6 (2018) ADSCrossRefGoogle Scholar
  18. 18.
    R. Kengne, R. Tchitnga, A. Mezatio, A. Fomethe, G. Litak, Eur. Phys. J. B 90, 88 (2017) ADSCrossRefGoogle Scholar
  19. 19.
    L.O. Chua, M. Komuro, T. Matsumoto, IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 33, 1073 (1986) ADSGoogle Scholar
  20. 20.
    T. Hartley, C. Lorenzo, H. Qammer, IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 42, 485 (1995) CrossRefGoogle Scholar
  21. 21.
    C.P. Li, W.H. Deng, D. Xu, Physica A 360, 171 (2006) ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    D. Cafagna, G. Grassi, Int. J. Bifurcat. Chaos 18, 615 (2008) CrossRefGoogle Scholar
  23. 23.
    Z. Chaoxia, Y. Simin, Chaos Soliton. Fract. 44, 845 (2011) CrossRefGoogle Scholar
  24. 24.
    C. Li, G. Chen, Chaos Soliton. Fract. 22, 540 (2004) ADSGoogle Scholar
  25. 25.
    R. Kengne, R. Tchitnga, K.A. Tchagna, A. Fomethe, J. Eng. Sci. Technol. Rev. 6, 24 (2013) CrossRefGoogle Scholar
  26. 26.
    C. Donato, G. Giuseppe, Nonlinear Dyn. 70, 1185 (2012) CrossRefGoogle Scholar
  27. 27.
    R.M. Nguimdo, R. Tchitnga, P. Woafo, Chaos 23, 043122 (2013) ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    R. Tchitnga, H.B. Fotsin, B. Nana, P.H. Louodop, P. Woafo, Chaos Soliton. Fract. 45, 306 (2012) ADSCrossRefGoogle Scholar
  29. 29.
    J.G. Freire, J.A.C. Gallas, Chaos Soliton. Fract. 59, 129 (2014) ADSCrossRefGoogle Scholar
  30. 30.
    J.G. Freire, R. Meucci, F.T. Arecchi, J.A.C. Gallas, Chaos 25, 097607 (2015) ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    J.G. Freire, C. Cabeza, A.C. Marti, T.P. Oschel, J.A.C. Gallas, Eur. Phys. J. Special Topics 223, 2857 (2014) ADSCrossRefGoogle Scholar
  32. 32.
    M.R. Gallas, J.A.C. Gallas, Chaos 25, 064603 (2015) ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    M.R. Gallas, J.A.C. Gallas, Eur. Phys. J. Special Topics 223, 2131 (2014) ADSCrossRefGoogle Scholar
  34. 34.
    L. Junges, J.A.C. Gallas, Phys. Lett. A 376, 2109 (2012) ADSCrossRefGoogle Scholar
  35. 35.
    L. Junges, T.P. Oschel, J.A.C. Gallas, Eur. Phys. J. D 67, 149 (2013) ADSCrossRefGoogle Scholar
  36. 36.
    J. Guan, Optik 127, 4211 (2016) ADSCrossRefGoogle Scholar
  37. 37.
    A. Soukkou, A. Boukabou, S. Leulmi, Optik 127, 5070 (2016) ADSCrossRefGoogle Scholar
  38. 38.
    H. Xi, Y. Li, X. Huang, Optik 126, 5346 (2015) ADSCrossRefGoogle Scholar
  39. 39.
    R. Li, W. Li, Optik 126, 2965 (2015) ADSCrossRefGoogle Scholar
  40. 40.
    G.C. Wu, D. Baleanu, L.L. Huang, Appl. Math. Lett. 82, 71 (2018) MathSciNetCrossRefGoogle Scholar
  41. 41.
    G.C. Wu, D. Baleanu, H.P. Xie, F.L. Chen, Physica A 460, 374 (2016) ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    W. Xingyuan, Z. Xiaopeng, M. Chao, Nonlinear Dyn. 69, 511 (2012) CrossRefGoogle Scholar
  43. 43.
    R. Kengne, R. Tchitnga, K.A. Tchagna, T.A. Nzeusseu, A. Fomethe, J. Chaos 2013, 839038 (2013) CrossRefGoogle Scholar
  44. 44.
    W. Sha, Y. Yongguang, D. Miao, Physica A 389, 4981 (2010) CrossRefGoogle Scholar
  45. 45.
    D. Hongyue, Chaos Soliton. Fract. 44, 510 (2011) CrossRefGoogle Scholar
  46. 46.
    X.J. Wu, H.T. Lu, Chin. Phys. B 19, 070511 (2010) ADSCrossRefGoogle Scholar
  47. 47.
    Y. Tang, J. Fang, Commun. Nonlinear Sci. Numer. Simul. 15, 401 (2010) ADSMathSciNetCrossRefGoogle Scholar
  48. 48.
    Y. Tang, Z. Wang, J. Fang, Chaos 19, 013112 (2009) ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Z. Darui, L. Ling, L. Chongxin, Math. Probl. Eng. 2014, 936985 (2014) Google Scholar
  50. 50.
    W.W. Yu, G. Chen, J.H. Lu, Automatica 45, 429 (2009) CrossRefGoogle Scholar
  51. 51.
    Y. Chai, L.P. Chen, R.C. Wu, J. Sun, Physica A 391, 5746 (2012) ADSMathSciNetCrossRefGoogle Scholar
  52. 52.
    J. Zhou, J.A. Lu, J.H. Lu, Automatica 44, 996 (2008) CrossRefGoogle Scholar
  53. 53.
    H. Fotsin, S. Bowong, Chaos Soliton. Fract. 27, 822 (2006) ADSCrossRefGoogle Scholar
  54. 54.
    I. Boiko, L. Fridman, R. Iriarte, A. Pisano, E. Usai, Automatica 42, 833 (2006) CrossRefGoogle Scholar
  55. 55.
    J. Slotine, W. Li, Applied Nonlinear Control (Prentice Hall, New Jersey, 1991) Google Scholar
  56. 56.
    J. Chen, C. Li, X. Yang, Neurocomputing 313, 324 (2018) CrossRefGoogle Scholar
  57. 57.
    J. Fei, C. Lu, J. Franklin Inst. 355, 2369 (2018) MathSciNetCrossRefGoogle Scholar
  58. 58.
    S.H. Hosseinnia, R. Ghaderi, A.N. Ranjbar, M. Mahmoudian, S. Momani, Comput. Math. Appl. 59, 1637 (2010) MathSciNetCrossRefGoogle Scholar
  59. 59.
    H.L. Li, J. Cao, H. Jiang, A. Alsaedi, J. Franklin Inst. 335, 5771 (2018) CrossRefGoogle Scholar
  60. 60.
    A. Syta, G. Litak, S. Lenci, M. Scheffler, Chaos 24, 013107 (2014) ADSMathSciNetCrossRefGoogle Scholar
  61. 61.
    M.A. Duarte-Mermoud, N. Aguila-Camacho, J.A. Gallegos, R. Castro-Linares, Commun. Nonlinear Sci. Numer. Simul. 22, 650 (2015) ADSMathSciNetCrossRefGoogle Scholar
  62. 62.
    A. Sharma, M.D. Shrimali, A. Prasad, R. Ramaswamy, U. Feudel, Phys. Rev. E 84, 016226 (2011) ADSCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://doi.org/creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  • Romanic Kengne
    • 1
    • 2
  • Robert Tchitnga
    • 1
    • 2
    • 3
  • Alain Kammogne Soup Tewa
    • 2
  • Grzegorz Litak
    • 4
    • 5
    Email author
  • Anaclet Fomethe
    • 6
  • Chunlai Li
    • 7
  1. 1.Research Group on Experimental and Applied Physics for Sustainable Development, Faculty of Science, Department of Physics, University of DschangDschangCameroon
  2. 2.Laboratory of Electronics and Signal Processing Faculty of Science, Department of Physics, University of DschangDschangCameroon
  3. 3.Institute of Surface Chemistry and Catalysis, University of UlmUlmGermany
  4. 4.Lublin University of Technology, Faculty of Mechanical EngineeringLublinPoland
  5. 5.AGH University of Science and Technology, Faculty of Mechanical Engineering and Robotics, Department of Process ControlKrakowPoland
  6. 6.Laboratoire de Mécanique et de Modélisation des Systèmes, L2MS, Department of Mathematics and Computer Science, Faculty of Science, University of DschangDschangCameroon
  7. 7.College of Physics and Electronics, Hunan Institute of Science and Technology YueyangHunanP.R. China

Personalised recommendations