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A new study of chaotic behavior and the existence of Feigenbaum’s constants in fractional-degree Yin–Yang Hénon maps

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Abstract

In this paper, we firstly develop fractional-degree Hénon maps with increasing and decreasing argument n. Yin and Yang are two fundamental opposites in Chinese philosophy. Yin represents the moon and is the decreasing, negative, historical, or feminine principle in nature, while Yang represents the sun and is the increasing, positive, contemporary, or masculine principle in nature. Chaos produced by increasing n is called Yang chaos, that by decreasing n Yin chaos, respectively. The simulation results show that chaos appears via positive Lyapunov exponents, bifurcation diagrams, and phase portraits. In order to examine the existence of chaotic behaviors in fractional-degree Yin–Yang Hénon maps, Feigenbaum’s constants are measured in this paper. It is found that the Feigenbaum’s constants in fractional-degree Yin–Yang Hénon maps are of great precision to the first and second Feigenbaum’s constants. A detailed analysis of the chaotic behaviors is also performed for the fractional-degree Hénon maps with increasing (Yang) and decreasing (Yin) argument n.

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References

  1. Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnéodo, A., Argoul, F., Elezgaray, J., Richetti, P.: Homoclinic chaos in chemical systems. Physica D 62, 134–169 (1993)

    Article  MATH  Google Scholar 

  4. Igeta, K., Ogawa, T.: Information dissipation in quantum-chaotic systems: computational view and measurement induction. Chaos Solitons Fractals 5, 1365–1379 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, H.K.: Global chaos synchronization of new chaotic systems via nonlinear control. Chaos Solitons Fractals 23, 1245–1251 (2005)

    MATH  Google Scholar 

  6. Lu, J., Wu, X., Lü, J.: Synchronization of a unified chaotic system and the application in secure communication. Phys. Lett. A 305, 365–370 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chua, L.O., Itah, M., Kosarev, L., Eckert, K.: Chaos synchronization in Chua’s circuits. J. Circuits Syst. Comput. 3, 93–108 (1993)

    Article  Google Scholar 

  8. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, Berlin (2003)

    MATH  Google Scholar 

  9. Bao, J., Yang, Q.: Complex dynamics in the stretch-twist-fold flow. Nonlinear Dyn. 61, 773–781 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wu, W., Chen, Z.: Hopf bifurcation and intermittent transition to hyperchaos in a novel strong four-dimensional hyperchaotic system. Nonlinear Dyn. 60, 615–630 (2010)

    Article  MATH  Google Scholar 

  11. Liu, Y., Pang, W.: Dynamics of the general Lorenz family. Nonlinear Dyn. 67, 1595–1611 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Śliwa, I., Grygiel, K.: Periodic orbits, basins of attraction and chaotic beats in two coupled Kerr oscillators. Nonlinear Dyn. 67, 755–765 (2012)

    Article  Google Scholar 

  13. Li, X., Ou, Q.: Dynamical properties and simulation of a new Lorenz-like chaotic system. Nonlinear Dyn. 65, 255–270 (2011)

    Article  MathSciNet  Google Scholar 

  14. Mirzaei, O., Yaghoobi, M., Irani, H.: A new image encryption method: parallel sub-image encryption with hyper chaos. Nonlinear Dyn. 67, 557–566 (2012)

    Article  MathSciNet  Google Scholar 

  15. Chen, C.S.: Optimal nonlinear observers for chaotic synchronization with message embedded. Nonlinear Dyn. 61, 623–632 (2010)

    Article  MATH  Google Scholar 

  16. Wang, D., Zhang, J.: Research and perspective of secure communication based on synchronization of chaos. J. Nav. Aeronaut. Eng. Inst. 21, 257–260 (2006)

    Google Scholar 

  17. Wang, X.Y., Yang, L., Liu, R., Kadir, A.: A chaotic image encryption algorithm based on perceptron model. Nonlinear Dyn. 62, 615–621 (2010)

    Article  MATH  Google Scholar 

  18. Ma, J., Li, A.B., Pu, Z.S., Yang, L.J., Wang, Y.-Z.: A time-varying hyperchaotic system and its realization in circuit. Nonlinear Dyn. 62, 535–541 (2010)

    Article  MATH  Google Scholar 

  19. Chen, Z., Ip, W.H., Chan, C.Y., Yung, K.L.: Two-level chaos-based video cryptosystem on H.263 codec. Nonlinear Dyn. 62, 647–664 (2010)

    Article  Google Scholar 

  20. Chen, G., Dong, X.: From Chaos to Order: Methodologies, Perspectives and Applications. World Scientific, Singapore (1988)

    Google Scholar 

  21. Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 50, 69–77 (1976)

    Article  MATH  Google Scholar 

  22. Lorenz, E.N.: Compound windows of the Hénon map. Physica D 237, 1689–1704 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sterling, D., Dullin, H.R., Meiss, J.D.: Homoclinic bifurcations for the Hénon map. Physica D 134, 153–184 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Feigenbaum, M.J.: Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19, 25–52 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  25. Feigenbaum, M.J.: Universal behavior in nonlinear systems. Los Alamos Sci. 1, 4–27 (1980)

    MathSciNet  Google Scholar 

  26. Goldfain, E.: Feigenbaum scaling, Cantorian space-time and the hierarchical structure of standard model parameters. Chaos Solitons Fractals 30, 324–331 (2006)

    Article  Google Scholar 

  27. San Martin, J.: Intermittency cascade. Chaos Solitons Fractals 32, 816–831 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Letellier, C., Bennoud, M., Martel, G.: Intermittency and period-doubling cascade on tori in a bimode laser model. Chaos Solitons Fractals 33, 782–794 (2007)

    Article  Google Scholar 

  29. Chen, H.K., Sheu, L.J., Tam, L.M., Lao, S.K.: A new finding of the existence of Feigenbaum’s constants in the fractional-order Chen–Lee system. Nonlinear Dyn. 68, 589–599 (2012)

    Article  MathSciNet  Google Scholar 

  30. Ge, Z.M., Li, S.Y.: Yang and Yin parameters in the Lorenz system. Nonlinear Dyn. 62, 105–117 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ho, C.Y., Chen, H.K., Ge, Z.M.: Design of PDC controllers by matrix reversibility for synchronization of Yin and Yang chaotic Takagi–Sugeno fuzzy Hénon maps. Abstract and Applied Analysis 11 pages (2012)

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Acknowledgements

The research was partially supported by a grant (NCS101-2221-E-164-008) from the National Science Council, R.O.C.

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Authors and Affiliations

  1. Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, 300, Taiwan

    Chun-Yen Ho & Zheng-Ming Ge

  2. Department of Mechanical Engineering, Hsiuping University of Science and Technology, 11 Gongye Rd., Dali Dist., Taichung, 412-80, Taiwan

    Hsien-Keng Chen

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  1. Chun-Yen Ho
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  2. Hsien-Keng Chen
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  3. Zheng-Ming Ge
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Correspondence to Zheng-Ming Ge.

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Ho, CY., Chen, HK. & Ge, ZM. A new study of chaotic behavior and the existence of Feigenbaum’s constants in fractional-degree Yin–Yang Hénon maps. Nonlinear Dyn 74, 439–453 (2013). https://doi.org/10.1007/s11071-013-0981-x

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