Advertisement

Ultimate boundary estimation and topological horseshoe analysis on a parallel 4D hyperchaotic system with any number of attractors and its multi-scroll

  • Enzeng DongEmail author
  • Zhijun Zhang
  • Mingfeng Yuan
  • Yuehui Ji
  • Xuesong Zhou
  • Zenghui Wang
Original Paper
  • 59 Downloads

Abstract

This paper constructs a new four-dimensional autonomous hyperchaotic system with complex dynamic behaviors, and its boundary is estimated based on the proposed method and the optimization idea. Inspired by the “parallel universe theory,” a trigonometric function is used to do coordinate transformation of the original new system to generate any number of attractors. Simulation results show that there are infinite equilibriums in the transformed system. Compared with the original new system, the transformed system is more sensitive to the initial values. Based on the estimated boundary of the original system, the boundary of transformed system could be obtained. To verify the existence of chaos of the transformed system, the topology horseshoe of the system is investigated. The positive topological entropy of the transformed system verifies the existence of hyperchaos. Furthermore, selecting proper parameter values, the transformed system shows a multi-scroll attractor. Applying multi-variable trigonometric transformation can also induce any number of attractors and multi-scroll phenomenon in multi-dimension, which is an interesting phenomenon.

Keywords

Hyperchaotic system Initial sensitivity Estimated boundary Any number of attractors Topological horseshoe Multi-scroll 

Notes

Acknowledgements

This work was partially supported by the Natural Science Foundation of China under Grant (Nos. 61603274 and 61502340), the Foundation of the Application Base and Frontier Technology Research Project of Tianjin (No. 15JCYBJC51800), South African National Research Foundation Grants (Nos. 112108 & 112142), South African National Research Foundation Incentive Grant (No. 114911) and Tertiary Education Support Programme (TESP) of South African ESKOM.

Compliance with ethical standards

Conflicts of interest

The authors declare no conflict of interest.

References

  1. 1.
    Orue, A.B., Alvarea, G., Pastor, G., Romera, M., Montoya, F., Li, S.: A new parameter determination method for some double-scroll chaotic systems and its applications to chaotic cryptanalysis. Common Nonlinear Sci. 15, 3471–3483 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Szolnoki, A., Mobilia, M., Jiang, L.L., Szczesny, B., Rucklidge, A.M.: Cyclic dominance in evolutionary games: a review. J. R. Soc. Interface 11, 100 (2014)CrossRefGoogle Scholar
  3. 3.
    Zhang, S., Gao, T.G.: A coding and substitution frame based on hyper-chaotic system for secure communication. Nonlinear Dyn. 84, 833–849 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Carroll, T.L.: Chaos for low probability of detection communications. Chaos Solitons Fractals 103, 238–245 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dimassi, H., Loria, A.: Adaptive unknown-input observers-based synchronization of chaotic systems for telecommunication. Circuits Syst. 58, 800–812 (2011)MathSciNetGoogle Scholar
  6. 6.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Phys. D 16, 285–317 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ai-Sawalha, M.M., Ai-Dababseh, A.F.: Nonlinear anti-synchronization of two hyperchaotic systems. Appl. Math. Sci. 5, 1849–1856 (2011)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bai, L., Zhang, G.: Nontrivial solutions for a nonlinear discrete elliptic equation with periodic boundary conditions. Appl. Math. Comput. 210, 321–333 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kaddoum, G., Lawrance, A.J., Chargé, P., Roviras, D.: Theory and computation: chaos communication performance. Circuits Syst. Signal Process 30, 185–208 (2011)CrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, C.C., Conejero, J.A., Kostic, M., Marina, M.A.: Dynamics on binary relations over topological spaces. Symmetry 10, 211 (2018)CrossRefGoogle Scholar
  11. 11.
    Sprott, J.C.: A new class of chaotic circuit. Phys. Lett. A 266, 19–23 (2000)CrossRefGoogle Scholar
  12. 12.
    Trigeassou, J.C., Maamri, N., Oustaloup, A.: Lyapunov stability of commensurate fractional order systems: a physical interpretation. Nonlinear Dyn. 11, 051007 (2016)CrossRefGoogle Scholar
  13. 13.
    Nik, H.S., Golchaman, M.: Chaos control of a bounded 4D chaotic system. Neural Comput. Appl. 25, 683–692 (2014)CrossRefGoogle Scholar
  14. 14.
    Campagnolo, A., Berto, F., Lazzarin, P.: The effects of different boundary conditions on three-dimensional cracked discs under anti-plane loading. Eur. J. Mech. A/Solids 50, 76–86 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cabada, A., Hamdi, Z.: Nonlinear fractional differential equations with integral boundary value conditions. Appl. Math. Comput. 228, 251–257 (2014)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Zhou, L.L., Chen, Z.Q., Wang, J.Z., Zhang, Q.: Local bifurcation analysis and global dynamics estimation of a novel 4-dimensional hyperchaotic system. Int. J. Bifurc. Chaos 27, 1750021 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Das, S., Pan, I., Das, S.: Effect of random parameter switching on commensurate fractional order chaotic systems. Chaos Solitons Fractals 91, 157–173 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wang, P., Li, D.M., Wu, X.Q., Yu, X.H.: Ultimate bound estimation of A class of high dimensional quadratic autonomous dynamical systems. Int. J. Bifurc. Chaos 21, 2679–2694 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Qi, G.Y., Zhang, J.F.: Energy cycle and bound of QI chaotic system. Chaos Solitons Fractals 99, 7–15 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ahmad, B., Ntouyas, S.K., Alsaedi, A.: On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions. Chaos Solitons Fractals 83, 234–241 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wang, X., Chen, G.R.: Constructing a chaotic system with any number of equilibria. Nonlinear Dyn. 71, 429–436 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Wei, Z.C.: Dynamical behaviors of a chaotic system with no equilibria. Phys. Lett. A 376, 102–108 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhang, X.: Constructing a chaotic system with any number of attractors. Int. J. Bifurc. Chaos 27, 1750118 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zhang, X., Chen, G.R.: Constructing an autonomous system with infinitely many chaotic attractors. Chaos 27, 071101 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Li, Q.D., Yang, S.: Research progress of chaotic dynamics based on topological horseshoe. J. Dyn. Control 10, 293–296 (2012)Google Scholar
  26. 26.
    Kennedy, J., Kocak, J.S., Yorke, J.A.: A chaos lemma. Am. Math Month. 108, 411–423 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical System and Chaos, 200–253. Springer, New York (2013)Google Scholar
  28. 28.
    Wang, Z.L., Zhou, L.L., Chen, Z.Q., Wang, J.Z.: Local bifurcation analysis and topological horseshoe of a 4D hyper-chaotic system. Nonlinear Dyn. 83, 2055–2066 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Li, Q.D., Zhang, L., Yang, F.: An algorithm to automatically detect the Smale horseshoes. Discret. Dyn. Nat. Soc. 1026, 726–737 (2012)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Lakshmi, B.: Chaotic dynamics in nonlinear theory, 29–53. Springer, India (2014)zbMATHGoogle Scholar
  31. 31.
    Yang, X.S.: Topological horseshoes and computer assisted verification of chaotic dynamics. Int. J. Circuit Theory Appl. 19, 1127–1145 (2009)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Qi, G.Y., Liang, X.Y.: Mechanical analysis of QI four-wing chaotic system. Nonlinear Dyn. 86, 1095–106 (2016)CrossRefGoogle Scholar
  33. 33.
    Pham, V.-T., Afari, S., Volos, C., Kapitaniak, T.: Different families of hidden attractors in a new chaotic system with variable equilibrium. Int. J. Bifurc. Chaos 27, 1750138 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Bao, B.-C., Jiang, P., Xu, Q., Chen, M.: Hidden attractors in a practical Chua’s circuit based on a modified Chua’s diode. Electron. Lett. 52, 23–25 (2015)CrossRefGoogle Scholar
  35. 35.
    Danca, M.F., Kuznetsov, N., Chen, G.R.: Unusual dynamics and hidden attractors of the Rabinovich–Fabrikant system. Nonlinear Dyn. 10, 1007 (2016)Google Scholar
  36. 36.
    Muñoz-Pacheco, J.M., Zambrano-Serrano, E., Félix-Beltrán, O., Gómez-Pavón, L.C., Luis-Ramos, A.: Synchronization of PWL function-based 2D and 3D multi-scroll chaotic systems. Nonlinear Dyn. 70, 163–1643 (2012)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Hu, X.Y., Liu, C.X., Liu, L., Yao, Y.P., Zheng, G.C.: Multi-scroll hidden attractors and multi-wing hidden attractors in a 5-dimensional memristive system. Chin. Phys. B 11, 110502 (2017)CrossRefGoogle Scholar
  38. 38.
    Ma, J., Wu, X.Y., Chu, R.T., Zhang, L.P.: Selection of multi-scroll attractors in Jerk circuits and their verification using Pspice. Nonlinear Dyn. 76, 1951–62 (2014)CrossRefGoogle Scholar
  39. 39.
    Hu, X., Liu, C., Liu, L.: Multi-scroll hidden attractors in improved Sprott A system. Nonlinear Dyn. 86, 1725–1734 (2016)CrossRefGoogle Scholar
  40. 40.
    Ai, X.X., Sun, K.H., He, S.B., Wang, H.H.: Design of grid multiscroll chaotic attractors via transformations. Int. J. Bifurc. Chaos 25, 1530027 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
  42. 42.
    Li, Q.D., Yang, X.S.: A simple method for finding topological horseshoes. Int. J. Bifurc. Chaos 20, 467–478 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Dong, E.Z., Yuan, M.F., Zhang, C., Tong, J.G.: Topological horseshoe analysis, ultimate boundary estimations of a new 4D Hyperchaotic system and its FPGA implementation. Int J. Bifurc. Chaos 28, 1850081 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Tianjin Key Laboratory For Control Theory and Applications in Complicated SystemsTianjin University of TechnologyTianjinChina
  2. 2.Department of Electrical and Mining EngineeringUniversity of South AfricaFloridaSouth Africa

Personalised recommendations