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Hidden attractors in a chaotic system with an exponential nonlinear term

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Abstract

Studying systems with hidden attractors is new attractive research direction because of its practical and threoretical importance. A novel system with an exponential nonlinear term, which can exhibit hidden attractors, is proposed in this work. Although new system possesses no equilibrium points, it displays rich dynamical behaviors, like chaos. By calculating Lyapunov exponents and bifurcation diagram, the dynamical behaviors of such system are discovered. Moreover, two important features of a chaotic system, the possibility of synchronization and the feasibility of the theoretical model, are also presented by introducing an adaptive synchronization scheme and designing a digital hardware platform-based emulator.

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References

  1. L. Pecora, T.L. Carroll, Phys. Rev. Lett. 64, 821 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, C.S. Zhou, Phys. Rep. 366, 1 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  3. S.H. Strogatz, Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Perseus Books, Massachusetts, US, 1994)

  4. E.N. Lorenz, J. Atmos. Sci. 20, 130 (1963)

    Article  ADS  Google Scholar 

  5. O.E. Rossler, Phys. Lett. A 57, 397 (1976)

    Article  ADS  Google Scholar 

  6. T. Kapitaniak, J. Sound Vibr. 102, 440 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  7. J.C. Sprott, Phys. Rev. E 50, R647 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  8. S. Vaidyanathan, Eur. Phys. J. Special Topics 223, 1519 (2014)

    Article  ADS  Google Scholar 

  9. Z. Wei, Phys. Lett. A 376, 102 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. S. Jafari, J.C. Sprott, S.M.R.H. Golpayegani, Phys. Lett. A 377, 699 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  11. X. Wang, G. Chen, Commun. Nonlinear Sci. Numer. Simul. 17, 1264 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  12. M. Molaie, S. Jafari, J.C. Sprott, S. Golpayegani, Int. J. Bifur. Chaos 23, 1350188 (2013)

    Article  Google Scholar 

  13. L.P. Shilnikov, Sov. Math. Docklady 6, 163 (1965)

    Google Scholar 

  14. G.A. Leonov, N.V. Kuznetsov, O.A. Kuznetsova, S.M. Seldedzhi, V.I. Vagaitsev, Trans. Syst. Contr. 6, 54 (2011)

    Google Scholar 

  15. G.A. Leonov, N.V. Kuznetsov, V.I. Vagaitsev, Physica D 241, 1482 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. G.A. Leonov, N.V. Kuznetsov, Int. J. Bifur. Chaos 23, 1330002 (2013)

    Article  MathSciNet  Google Scholar 

  17. G.A. Leonov, N.V. Kuznetsov, IFAC. Proc. 18, 2494 (2011)

    Google Scholar 

  18. G.A. Leonov, N.V. Kuznetsov, Dokl. Math. 84, 475 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. G.A. Leonov, N.V. Kuznetsov, M.A. Kiseleva, E.P. Solovyeva, A.M. Zaretskiy, Nonlinear Dyn. 77, 277 (2014)

    Article  Google Scholar 

  20. X. Wang, G. Chen, Nonlinear Dyn. 71, 429 (2013)

    Article  Google Scholar 

  21. S.T. Kingni, S. Jafari, H. Simo, P. Woafo, Eur. Phys. J. Plus 129, 76 (2014)

    Article  Google Scholar 

  22. V.T. Pham, C.K. Volos, S. Jafari, Z. Wei, X. Wang, Int. J. Bifur. Chaos 24, 1450073 (2014)

    Article  MathSciNet  Google Scholar 

  23. S.K. Lao, Y. Shekofteh, S. Jafari, J.C. Sprott, Int. J. Bifur. Chaos 24, 1450010 (2014)

    Article  MathSciNet  Google Scholar 

  24. K.H. Sun, J.C. Sprott, Int. J. Nonlinear Sci. Num. Simu. 10, 1443 (2009)

    Google Scholar 

  25. B. Munmuangsaen, B. Srisuchinwong, J.C. Sprott, Phys. Lett. A 375, 1445 (2011)

    Article  ADS  MATH  Google Scholar 

  26. J.C. Sprott, IEEE Trans. Circuits Syst.-II: Exp. Briefs 58, 240 (2011)

    Article  Google Scholar 

  27. F. Yu, C. Wang, Eng. Tech. Appl. Sci. Research 2, 209 (2012)

    Google Scholar 

  28. I. Ahmed, C. Mu, F. Zhang, Int. J. Eng. Anal. Applications 5, 27 (2014)

    Google Scholar 

  29. A.R. Sahab, M.T. Ziabari, M.R. Modabbernia, Adv. Diff. Equ. 2012, 1 (2012)

    Article  Google Scholar 

  30. Z. Wei, Q. Yang, Nonl. Anal. Real World Appl. 12, 106 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Z. Wei, J. Wang, Optoelectron. Adv. Mater. Rapid Comm. 6, 926 (2012)

    Google Scholar 

  32. G. Chen, T. Ueta, Int. J. Bifur. Chaos 9, 1465 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  33. S. Jafari, J.C. Sprott, Chaos Solitons Fractals 57, 79 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  34. O.E. Rossler, Phys. Lett. A 71, 155 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  35. S. Vaidyanathan, I. Pehlivan, Math. Comp. Modelling 55, 1904 (2012)

    Article  MATH  Google Scholar 

  36. L. Fortuna, M. Frasca, Chaos 17, 043118 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  37. M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 78, 4193 (1997)

    Article  ADS  Google Scholar 

  38. P. Woafo, H.G.E. Kadji, Phys. Rev. E 69, 046206 (2004)

    Article  ADS  Google Scholar 

  39. A. Akopov, V. Astakhov, T. Vadiasova, A. Shabunin, T. Kapitaniak, Phys. Lett. A 334, 169 (2005)

    Article  ADS  MATH  Google Scholar 

  40. A. Stefanski, P. Perlikowski, T. Kapitaniak, Phys. Rev. E 75, 016210 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  41. P. Perlikowski, A. Stefanski, T. Kapitaniak, Phys. Rev. E 77, 048201 (2008)

    Article  ADS  Google Scholar 

  42. P. Kuzma, M. Kapitaniak, T. Kapitaniak, J. Theor. Appl. Mech. 52, 281 (2014)

    Google Scholar 

  43. R. Karthikeyan, S. Vaidyanathan, J. Electrical Eng. 65, 97 (2014)

    Article  Google Scholar 

  44. S. Vaidyanathan, C. Volos, V.T. Pham, K. Madhavan, B.A. Idowo, Archives Cont. Sci. 33, 257 (2014)

    Google Scholar 

  45. O. Morgul, E. Solak, Phys. Rev. E 54, 4803 (1996)

    Article  ADS  Google Scholar 

  46. O. Morgul, E. Solak, Int. J. Bifur. Chaos 7, 1307 (1997)

    Article  MathSciNet  Google Scholar 

  47. H. Nijmeijer, I. Mareels, IEEE Trans. Circuits Syst.–I 44, 882 (1997)

    Article  MathSciNet  Google Scholar 

  48. J. Mata-Machuca, R. Martinez-Guerra, R. Aguilar-Lopez, C. Aguilar-Ibanez, Commun. Nonlinear Sci. Numer. Simul. 17, 1706 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. R. Aguilar-Lopez, R. Martinez-Guerra, C. Perez-Pinacho, Eur. Phys. J. Special Topics 223, 1541 (2014)

    Article  ADS  Google Scholar 

  50. H.K. Khalil, Nonlinear Systems, 2nd edn. (Prentice Hall, New Jersey, USA, 2001)

  51. L. Fortuna, M. Frasca, M.G. Xibilia, Chua’s Circuit Implementation: Yesterday, Today and Tomorrow (World Scientific, Singapore, 2009)

  52. Q. Ding, J. Pang, J. Fang, X.U. Peng, Int. J. Innov. Comput. Inf. Control 3, 449 (2007)

    Google Scholar 

  53. G.Y. Wang, X.L. Bao, Z.L. Wang, Chin. Phys. B 17, 3596 (2008)

    Article  ADS  Google Scholar 

  54. M.S. Azzaz, C. Tanougast, S. Sadoudi, R. Fellah, A. Dandache, Commun. Nonlinear Sci. Numer. Simul. 18, 1792 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  55. S. Sadoudi, M.S. Azzaz, M. Djeddou, M. Benssalah, Int. J. Nonlinear Sci. 7, 467 (2009)

    MathSciNet  Google Scholar 

  56. I. Koyuncu, A.T. Ozcerit, I. Pehlivan, Nonlinear Dyn. 77, 49 (2014)

    Article  MathSciNet  Google Scholar 

  57. D. Valli, B. Muthuswamy, S. Banerjee, M.R.K. Ariffin, A.W.A. Wahad, K. Ganesan, C.K. Subramaniam, J. Kurths, Eur. Phys. J. Special Topics 223, 1465 (2014)

    Article  ADS  Google Scholar 

  58. V.T. Pham, L. Fortuna, M. Frasca, Nonlinear Dyn. 67, 345 (2012)

    Article  Google Scholar 

  59. E. Monmansson, M. Cirstea, IEEE Trans. Ind. Electron. 54, 1824 (2007)

    Article  Google Scholar 

  60. D. Harris, S. Harris, Digital Design and Computer Architecture, 2nd edn. (Morgan Kaufmann, USA, 2012)

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Authors and Affiliations

  1. School of Electronics and Telecommunications, Hanoi University of Science and Technology, 01 Dai Co Viet, Hanoi, Vietnam

    V.-T. Pham

  2. Research and Development Centre, Vel Tech University, Avadi, Chennai, 600062, Tamil Nadu, India

    S. Vaidyanathan

  3. Physics Department, Aristotle University of Thessaloniki, 54124, Thessaloniki, Greece

    C. K. Volos

  4. Biomedical Engineering Department, Amirkabir University of Technology, Tehran, 15875-4413, Iran

    S. Jafari

Authors
  1. V.-T. Pham
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  2. S. Vaidyanathan
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  3. C. K. Volos
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  4. S. Jafari
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Correspondence to V.-T. Pham.

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Pham, VT., Vaidyanathan, S., Volos, C.K. et al. Hidden attractors in a chaotic system with an exponential nonlinear term. Eur. Phys. J. Spec. Top. 224, 1507–1517 (2015). https://doi.org/10.1140/epjst/e2015-02476-9

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  • DOI: https://doi.org/10.1140/epjst/e2015-02476-9

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