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Coexistence of attractors in autonomous Van der Pol–Duffing jerk oscillator: Analysis, chaos control and synchronisation in its fractional-order form

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Abstract

In this paper, a Van der Pol–Duffing (VdPD) jerk oscillator is designed. The proposed VdPD jerk oscillator is built by converting the autonomous two-dimensional VdPD oscillator to a jerk oscillator. Dynamical behaviours of the proposed VdPD jerk oscillator are investigated analytically, numerically and analogically. The numerical results indicate that the proposed VdPD jerk oscillator displays chaotic oscillations, symmetrical bifurcations and coexisting attractors. The physical existence of the chaotic behaviour found in the proposed VdPD jerk oscillator is verified by using Orcad-PSpice software. A good qualitative agreement is shown between the numerical simulations and the PSpice results. Moreover, the fractional-order form of the proposed VdPD jerk oscillator is studied using stability theorem of fractional-order systems and numerical simulations. It is found that chaos, periodic oscillations and coexistence of attractors exist in the fractional-order form of the proposed jerk oscillator with order less than three. The effect of fractional-order derivative on controlling chaos is illustrated. It is shown that chaos control is achieved in fractional-order form of the proposed VdPD jerk oscillator only for the values of linear controller used. Finally, the problem of drive–response synchronisation of the fractional-order form of the chaotic proposed VdPD jerk oscillators is considered using active control technique.

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References

  1. W J Rudowski and J Szemplinska-Stupnicka, J. Sound Vib. 199, 165 (1997)

    Article  ADS  Google Scholar 

  2. G J Fodjouong, H B Fotsin and P Woafo, Phys. Scr. 75, 638 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  3. H B Fotsin, S Bowong and J Daafouz, Chaos Solitons Fractals 26, 215 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  4. H B Fotsin and P Woafo, Chaos Solitons Fractals 24, 1363 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  5. U E Vincent, R K Odunaike, J A Laoye and A A Gbindinninuola, J. Control Theory Appl. 9, 141 (2011)

    Article  MathSciNet  Google Scholar 

  6. T Kapitaniak, Chaos for engineers: Theory, applications and control (Springer, New York, 1998)

    Book  MATH  Google Scholar 

  7. A Maccari, J. Sound Vib. 317, 20 (2008)

    Article  ADS  Google Scholar 

  8. J Cui, J Liang and Z Lin, Phys. Scr. 91, 015201 (2016)

    Article  ADS  Google Scholar 

  9. D Xue, Y Q Chen and D P Atherton, Linear feedback control analysis and design with Matlab (Siam, Bangkok, 2007)

    Book  MATH  Google Scholar 

  10. A K Jonscher, Dielectric relaxation in solids (Chelsea Dielectric Press, London, 1993)

    Google Scholar 

  11. I Schafer and K Kruger, J. Phys. D: Appl. Phys. 41, 1 (2008)

    Google Scholar 

  12. S Faraji and M S Tavazoei, Cent. Eur. J. Phys. 11, 836 (2013)

    Google Scholar 

  13. G S Mbouna Ngueuteu and P Woafo, Mech. Res. Commun. 46, 20 (2012)

    Article  Google Scholar 

  14. L Preda, M Mihailescu and A Preda, UPB Sci. Bull. Ser. A 71, 11 (2009)

    Google Scholar 

  15. I Grigorenko and E Grigorenko, Phys. Rev. Lett. 91, 034101 (2003)

    Article  ADS  Google Scholar 

  16. S P Wang, S K Lao, H K Chen, J H Chen and S Y Chen, Int. J. Bifurc. Chaos 23, 1350030 (2013)

    Article  Google Scholar 

  17. A E Matouk, Commun. Nonlinear Sci. Numer. Simul. 16, 975 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  18. S T Kingni, G S Mbouna Ngueuteu and P Woafo, Nonlinear Dyn. 76, 1169 (2014)

    Article  Google Scholar 

  19. S T Kingni, B Nana, G S Mbouna Ngueuteu and P Woafo, Chaos Solitons Fractals 71, 29 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  20. S He, K Sun and S Banerjee, Eur. Phys. J. Plus. 131, 254 (2016)

    Article  Google Scholar 

  21. M S Benitez, L A Zuppa and R J R Guerra, IEICE Trans. Fundam. 89, 1088 (2006)

    Article  Google Scholar 

  22. J M Malasoma, Phys. Lett. A 264, 383 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  23. H P W Gottlieb, Am. J. Phys. 64, 525 (1996)

    Article  ADS  Google Scholar 

  24. J C Sprott, IEEE Trans. Circuits Syst. II Express Briefs 58, 240 (2011)

    Article  Google Scholar 

  25. L Acho, J Rolon and S A Benitez, WSEAS Trans. Circuits Syst. 3, 198 (2004)

    Google Scholar 

  26. P Louodop, M Kountchou, H B Fotsin and S Bowong, Nonlinear Dyn. 78, 597 (2014)

    Article  Google Scholar 

  27. J Kengne, Z T Njitacke and H B Fotsin, Nonlinear Dyn. 83, 751 (2016)

    Article  Google Scholar 

  28. V Kamdoum Tamba, H B Fotsin, J Kengne, E B Megam Ngouonkadi and P K Talla, Int. J. Dynam. Control, https://doi.org/10.1007/s40435-016-0223-4 (2016)

  29. J Kengne, J C Chedjou, M Kom, K Kyamakya and V Kamdoum Tamba, Nonlinear Dyn. 761119 (2014)

  30. J Kengne, Z T Njitacke, V Kamdoum Tamba and A Nguomkam Negou, Chaos 25, 103126 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  31. A N Pisarchik and U Feudel, Phys. Rep. 540, 167 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  32. A N Pisarchik, Phys. Rev. E 64, 046203 (2001)

    Article  ADS  Google Scholar 

  33. K Diethelm, N J Ford and D A Freed, Nonlinear Dyn. 29, 3 (2002)

    Article  Google Scholar 

  34. R Caponetto, R Dongola, L Fortuna and I Petras, World Sci. Ser. Nonlinear Sci. Ser. A 72, 3 (2010)

    Google Scholar 

  35. R Hilfer, Applications of fractional calculus in physics (World Scientific, New Jersey, 2001)

    MATH  Google Scholar 

  36. M S Tavazoei and M A Haeri, Phys. Lett. A 367, 102 (2007)

    Article  ADS  Google Scholar 

  37. T Wang and N Jia, Appl. Math. Comput. 218, 7231 (2012)

    MathSciNet  Google Scholar 

  38. Y Luo, Y Q Chen, H S Ahn and Y G Pi, Control Eng. Pract. 18, 1022 (2010)

    Article  Google Scholar 

  39. H Delavari, R Ghaderi, A Ranjbar and S Momani, Commun. Nonlinear Sci. Numer. Simul. 15, 963 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  40. A V J Razminia Majd and D Baleanu, Adv. Differ. Equ. 15, 1 (2011)

    Google Scholar 

  41. E Ahmed, A M A El-Sayed and H A El-Saka, Phys. Lett. A 358, 1 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  42. M Falahpoor, M Ataei and A Kiyoumarsi, Chaos Solitons Fractals 42, 1755 (2009)

    Article  ADS  Google Scholar 

  43. A Nourian and S Balochian, Pramana – J. Phys. 86, 1401 (2016)

    Article  ADS  Google Scholar 

  44. U E Vincent, Chaos Solitons Fractals 37, 1065 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  45. M L Hung, J S Lin, J J Yan and T L Liao, Chaos Solitons Fractals 35, 781 (2008)

    Article  ADS  Google Scholar 

  46. A Khan, D Khattar and N Prajapati, Pramana – J. Phys . 88, 40 (2017)

    Article  ADS  Google Scholar 

  47. S Bowong and F M Kakmeni, Chaos Solitons Fractals 21, 999 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  48. K Sebastian Sudheer and M Sabir, Pramana – J. Phys. 88, 47 (2017)

    Article  ADS  Google Scholar 

  49. H Targhvafard and G H Enjace, Commun. Nonlinear Sci. Numer. Simul. 16, 4079 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  50. M Srivastava, S P Ansari, S K Agrawal, S Das and A Y T Leung, Nonlinear Dyn. 76, 905 (2014)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Telecommunication and Network Engineering, IUT-Fotso Victor of Bandjoun, University of Dschang, P.O. Box 134, Bandjoun, Cameroon

    Victor Kamdoum Tamba

  2. Department of Mechanical and Electrical Engineering, Faculty of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon

    Sifeu Takougang Kingni

  3. Department of Physics, Higher Teacher Training College, University of Bamenda, P.O. Box 39, Bamenda, Cameroon

    Gaetan Fautso Kuiate

  4. Laboratory of Electronics and Signal Processing (LETS), Department of Physics, Faculty of Science, University of Dschang, P.O. Box 67, Dschang, Cameroon

    Victor Kamdoum Tamba & Hilaire Bertrand Fotsin

  5. Laboratory of Mechanics and Modelling of Physical Systems (L2MPS), Department of Physics, Faculty of Science, University of Dschang, P.O. Box 67, Dschang, Cameroon

    Pierre Kisito Talla

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  1. Victor Kamdoum Tamba
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  2. Sifeu Takougang Kingni
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  3. Gaetan Fautso Kuiate
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  4. Hilaire Bertrand Fotsin
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  5. Pierre Kisito Talla
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Correspondence to Victor Kamdoum Tamba.

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Tamba, V.K., Kingni, S.T., Kuiate, G.F. et al. Coexistence of attractors in autonomous Van der Pol–Duffing jerk oscillator: Analysis, chaos control and synchronisation in its fractional-order form. Pramana - J Phys 91, 12 (2018). https://doi.org/10.1007/s12043-018-1586-1

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  • DOI: https://doi.org/10.1007/s12043-018-1586-1

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