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A novel chaotic system without equilibria, with parachute and thumb shapes of Poincare map and its projective synchronisation

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Abstract

In this paper, a three-dimensional novel chaotic system and its projective synchronisation are investigated. The proposed chaotic system has no equilibria. The topological structure of proposed chaotic system is different form Lorenz, Rossler and Chen systems. Different qualitative and quantitative tools such as time series, phase plane, Poincare section, bifurcation plot, Lyapunov exponents, Lyapunov spectrum, and Lyapunov dimension are used to evidence the chaotic behaviour of the proposed system. Further, the projective synchronisation between the proposed chaotic systems is achieved using nonlinear active control. Active control laws are designed, by using sum of the relevant variables of the proposed chaotic systems, to ensure the convergence of error dynamics. The required global asymptotic stability condition is derived using Lyapunov stability theory. Simulation is done in MATLAB environment to verify the theoretical approach. Simulation results reveal that the objectives of the paper are achieved successfully.

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References

  1. G. Chen, X. Dong, in From chaos to order: methodologies, perspectives and applications (World Scientific, Singapore, 1998), pp. 311–387.

  2. P.P. Singh, K.M. Singh, B.K. Roy, Eur. Phys. J. Special Topics 227, 731 (2018).

    ADS  Google Scholar 

  3. P.P. Singh, B.K. Roy, Annu. Rev. Control 45, 152 (2018).

    MathSciNet  Google Scholar 

  4. S. Banerjee, L. Rondoni, M. Mitra, Applications of chaos and nonlinear dynamics in science and engineering (Springer, Berlin, 2015), Vol. 3, pp. 371–392.

    ADS  Google Scholar 

  5. S. Mukherjee, S.K. Palit, S. Banerjee, M.R.K. Ariffin, L. Rondoni, D.K. Bhattacharya, Physica A 439, 93 (2015).

    ADS  MathSciNet  Google Scholar 

  6. S. Banerjee, J. Kurths, Eur. Phys. J. Special Topics 223, 1441 (2014).

    ADS  Google Scholar 

  7. P.P. Singh, J.P. Singh, B.K. Roy, IETE J. Res. 63, 853 (2017).

    Google Scholar 

  8. P.P. Singh, J.P. Singh, B.K. Roy, Int. J. Control Theory Appl. 9, 171 (2016).

    Google Scholar 

  9. P.P. Singh, J.P. Singh, B.K. Roy, Int. J. Control Theory Appl. 9, 159 (2016).

    Google Scholar 

  10. P.P. Singh, J.P. Singh, B.K. Roy, Int. J. Control Theory Appl. 8, 995 (2015).

    Google Scholar 

  11. P.P. Singh, J.P. Singh, B.K. Roy, Res. Rev.: J. Phys. 3, 1 (2014).

    Google Scholar 

  12. P.P. Singh, J.P. Singh, B.K. Roy, Chaos Solitons Fractals 69, 31 (2014).

    ADS  MathSciNet  Google Scholar 

  13. J.P. Singh, B.K. Roy, S. Jafari, Chaos Solitons Fractals 106, 243 (2018).

    ADS  MathSciNet  Google Scholar 

  14. J.P. Singh, B.K. Roy, Optik 145, 209 (2017).

    ADS  Google Scholar 

  15. P.P. Singh, J.P. Singh, M. Borah, B.K. Roy, in 4th Int Conference on Advances in Control and Optimization of Dynamical Systems (ACODS) (2016), Vol. 49, p. 522.

    Google Scholar 

  16. S. Jafari, V.T. Pham, T. Kapitaniak, Int. J. Bifurc. Chaos 26, 1650031 (2016).

    Google Scholar 

  17. S. Jafari, J.C. Sprott, F. Nazarimehr, Eur. Phys. J. Special Topics 224, 1469 (2015).

    ADS  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  19. S. Jafari, J.C. Sprott, S. Golpayegani, Phys. Lett. A 337, 699 (2013).

    ADS  Google Scholar 

  20. G.A. Leonov, N.V. Kuznetsov, V.I. Vagaitsev, Phys. Lett. A 375, 2230 (2011).

    ADS  MathSciNet  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  22. Z. Wei, I. Moroz, J.C. Sprott, A. Akgul, W. Zhang, Chaos 27, 033101 (2017).

    ADS  MathSciNet  Google Scholar 

  23. Z. Wei, I. Moroz, J.C. Sprott, Z. Wang, W. Zhang, Int. J. Bifurc. Chaos 27, 1730008 (2017).

    Google Scholar 

  24. Z. Wei, P. Yu, W. Zhang, M. Yao, Nonlinear Dyn. 82, 131 (2015).

    Google Scholar 

  25. Z. Wei, W. Zhang, Int. J. Bifurc. Chaos 24, 1450127 (2014).

    Google Scholar 

  26. Z. Wei, W. Zhang, Z. Wang, M. Yao, Int. J. Bifurc. Chaos 25, 1550028 (2015).

    Google Scholar 

  27. E.N. Lorenz, J. Atoms. Sci. 20, 130 (1963).

    ADS  Google Scholar 

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

    Google Scholar 

  29. G. Chen, T. Ueta, Int. J. Bifurc. Chaos 9, 1465 (1999).

    Google Scholar 

  30. J.C. Sprott, Phys. Rev. E 50, 647 (1994).

    ADS  MathSciNet  Google Scholar 

  31. L. Shilnikov, Sov. Math. Dokl. 6, 163 (1965).

    Google Scholar 

  32. L. Shilnikov, Math USS-R-Sbornik 10, 91 (1970).

    Google Scholar 

  33. C.P. Silva, , IEEE Trans. Circ. Syst. 40, 657 (1993).

    Google Scholar 

  34. Q.G. Yang, Z.C. Wei, G.R. Chen, Int. J. Bifurc. Chaos 20, 1061 (2010).

    Google Scholar 

  35. Q.G. Yang, G.R. Chen, Int. J. Bifurc. Chaos 18, 1393 (2008).

    Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  37. Z. Wei, Y. Li, Int. J. Bifurc. Chaos 29, 1950095 (2019).

    Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  39. G. Leonov, N. Kuznetsov, M. Kiseleva, E. Solovyeva, A. Zaretskiy, Nonlinear Dyn. 77, 288 (2014).

    Google Scholar 

  40. G. Leonov, N. Kuznetsov, T. Mokaev, Commun. Nonlinear Sci. Numer. Simul. 28, 174 (2015).

    ADS  Google Scholar 

  41. G. Leonov, N. Kuznetsov, V. Vagaitsev, Physica D 241, 1486 (2010).

    Google Scholar 

  42. F. Nazarimehr, B. Saedi, S. Jafari, J.C. Sprott, Int. J. Bifurc. Chaos 27, 1750037 (2015).

    Google Scholar 

  43. S. Brezetskyi, D. Dudkowski, T. Kapitaniak, Eur. Phys. J. Special Topics 224, 1467 (2015).

    ADS  Google Scholar 

  44. T. Kapitaniak, G.A. Leonov, Eur. Phys. J. Special Topics 224, 1408 (2015).

    ADS  Google Scholar 

  45. S. Jafari, J. Sprott, S.M.R.H. Golpayegani, Phys. Lett. A 377, 702 (2013).

    ADS  Google Scholar 

  46. V.T. Pham, C. Volos, S. Jafari, T. Kapitaniak, Nonlinear Dyn. 87, 2010 (2017).

    Google Scholar 

  47. V.T. Pham, S.T. Kingni, C. Volos, S. Jafari, T. Kapitaniak, AEU Int. J. Electron. Commun. 78, 227 (2017).

    Google Scholar 

  48. V.T. Pham, A. Akgul, C. Volos, S. Jafari, T. Kapitaniak, AEU Int. J. Electron. Commun. 78, 140 (2017).

    Google Scholar 

  49. S. Jafari, J.C. Sprott, Chaos Solitons Fractals 57, 84 (2013).

    ADS  Google Scholar 

  50. S.T. Kingni, V.T. Pham, S. Jafari, P. Woafo, Chaos Solitons Fractals 99, 218 (2017).

    ADS  Google Scholar 

  51. K. Barati, S. Jafari, J.C. Sprott, V.T. Pham, Int. J. Bifurc. Chaos 26, 1630034 (2016).

    Google Scholar 

  52. V.T. Pham, S. Jafari, C. Volos, T. Gotthans, X. Wang, D.V. Hoang, Optik 130, 371 (2017).

    ADS  Google Scholar 

  53. V.T. Pham, S. Jafari, C. Volos, Optik 131, 349 (2017).

    ADS  Google Scholar 

  54. V.T. Pham, C. Volos, T. Kapitaniak, S. Jafari, X. Wang, Int. J. Electron. 13 (2017).

  55. S. Jafari, J.C. Sprott, M. Molaie, Int. J. Bifurc. Chaos 26, 1650098 (2016).

    Google Scholar 

  56. B. Bao, T. Jiang, G. Wang, P. Jin, H. Bao, M. Chen, Nonlinear Dyn. 89, 1157 (2017).

    Google Scholar 

  57. S. Jafari, J.C. Sprott, V.T. Pham, C. Volos, C. Li, Nonlinear Dyn. 86, 1358 (2016).

    Google Scholar 

  58. M. Molaie, S. Jafari, J.C. Sprott, S.M.R.H. Golpayegani, Int. J. Bifurc. Chaos 23, 1350188 (2013).

    Google Scholar 

  59. V.T. Pham, S. Jafari, T. Kapitaniak, C. Volos, S.T. Kingni, Int. J. Bifurc. Chaos 27, 1750053 (2017).

    Google Scholar 

  60. K. Rajagopal, S. Jafari, G. Laarem, Pramana 89, 92 (2017).

    ADS  Google Scholar 

  61. K. Rajagopal, A. Akgul, S. Jafari, A. Karthikeyan, I. Koyuncu, Chaos Solitons Fractals 103, 487 (2017).

    Google Scholar 

  62. V.T. Pham, X. Wang, S. Jafari, C. Volos, T. Kapitaniak, Int. J. Bifurc. Chaos 27, 1750097 (2017).

    Google Scholar 

  63. V.T. Pham, S. Jafari, C. Volos, T. Kapitaniak, Int. J. Bifurc. Chaos 27, 1750138 (2017).

    Google Scholar 

  64. Z. Wei, V.T. Pham, A.J.M. Khalaf, J. Kengne, S. Jafari, Int. J. Bifurc. Chaos 28, 1850085 (2018).

    Google Scholar 

  65. B. Blazejczyk-Okolewska, T. Kapitaniak, Chaos Solitons Fractals 9, 1439 (1998).

    ADS  Google Scholar 

  66. T. Kapitaniak, Eur. Phys. J. Special Topics 224, 1405 (2015).

    ADS  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  68. P. Jaros, P. Perlikowski, T. Kapitaniak, Eur. Phys. J. Special Topics 224, 1541 (2015).

    ADS  Google Scholar 

  69. P. Jaros, P. Perlikowski, T. Kapitaniak, Eur. Phys. J. Special Topics 225, 2623 (2016).

    ADS  Google Scholar 

  70. Q. Lai, L. Wang, Optik 127, 5400 (2016).

    ADS  Google Scholar 

  71. B.C. Bao, Q. Xu, H. Bao, M. Chen, Electron Lett. 52, 1008 (2016).

    Google Scholar 

  72. B.C. Bao, H. Bao, M. Chen, P. Jin, Q. Xu, Chaos Solitons Fractals 94, 102 (2017).

    ADS  MathSciNet  Google Scholar 

  73. J.C. Sprott, S. Jafari, A.J.M. Khalaf, T. Kapitaniak, Eur. Phys. J. Special Topics 226, 1979 (2017).

    ADS  Google Scholar 

  74. V. Kumar, E. Tiwari, V.K. Chauhan, P.P. Singh, Int. J. Eng. Technol. 7, 50 (2018).

    Google Scholar 

  75. P.P. Singh, J.P. Singh, B.K. Roy, IFAC Proc. 47, 287 (2014).

    Google Scholar 

  76. J.P. Singh, P.P. Singh, B.K. Roy, IFAC Proc. 47, 292 (2014).

    Google Scholar 

  77. P.P. Singh, J.P. Singh, B.K. Roy, IEEE International Conference on Communication and Signal Processing-ICCSP’14 (IEEE, 2014).

  78. L. Basnarkov, G.S. Duane, L. Kocarev, Chaos Solitons Fractals 59, 35 (2014).

    ADS  MathSciNet  Google Scholar 

  79. A. Hu, Z. Xu, Nonlinear Sci. Numer. Simul. 16, 3237 (2011).

    Google Scholar 

  80. P.P. Singh, B.K. Roy, Eur. Phys. J. Special Topics 228, 2197 (2019).

    ADS  Google Scholar 

  81. Y. Shi, P. Zhu, K. Qin, Neurocomputing 123, 443 (2014).

    Google Scholar 

  82. J. Yu, C. Hu, H. Jiang, X. Fan, Neural Networks 49, 87 (2014).

    Google Scholar 

  83. P. Sarasu, V. Sundarapandian, Adv. Appl. Chaot. Syst. Stud. Comput. Intel. 636, 427 (2016).

    Google Scholar 

  84. P. Sarasu, V. Sundarapandian, Eur. J. Sci. Res. 72, 504 (2012).

    Google Scholar 

  85. P. Sarasu, V. Sundarapandian, Int. J. Soft. Comput. 7, 146 (2012).

    Google Scholar 

  86. S. Wang, X. Wang, Y. Zhou, Entropy 17, 7628 (2015).

    ADS  Google Scholar 

  87. M. Moghtadaei, M.R.H. Golpayegani, Sci. Iran. D 19, 733 (2012).

    Google Scholar 

  88. P.P. Singh, B.K. Roy, Eur. Phys. J. Special Topics 5, 861 (2020).

    Google Scholar 

  89. Y.A. Kuznetsov, Elements of applied bifurcation theory, 2nd ed. (Springer, New York, 1998).

  90. L. Yan, J. Jhang, X. Hou, Nonlinear systems of algebraic equations and mechanical theorem proving (Shanghai Scientific and Technological Education Publishing House, Shanghai, 1996).

  91. J. Kaplan, J. Yorke, in Functional differential equations and the approximation of fixed points, Lecture notes in mathematics (Springer, Berlin, 1979), pp. 204–227.

  92. J.E. Slotine, W. Li, in Applied nonlinear control (Prentice Hall Inc., Englewood Cliffs, NJ, 1991), pp. 100–154.

  93. J.E. Slotine, W. Li, in Applied nonlinear control (Prentice Hall Inc., Englewood Clis, NJ, 1991), pp. 276–311.

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Authors and Affiliations

  1. Department of Electrical Engineering, National Institute of Technology Meghalaya, 793003, Shillong, Meghalaya, India

    Piyush Pratap Singh

  2. Department of Electrical Engineering, National Institute of Technology Silchar, 788010, Silchar, Assam, India

    Binoy Krishna Roy

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  1. Piyush Pratap Singh
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  2. Binoy Krishna Roy
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Correspondence to Piyush Pratap Singh.

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Singh, P.P., Roy, B.K. A novel chaotic system without equilibria, with parachute and thumb shapes of Poincare map and its projective synchronisation. Eur. Phys. J. Spec. Top. 229, 1265–1278 (2020). https://doi.org/10.1140/epjst/e2020-900259-0

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