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On hidden twin attractors and bifurcation in the Chua’s circuit

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Abstract

Recently a new attractor, called hidden attractor, has been found in the well-known Chua’s circuit, whose basin of attraction does not contain neighborhood of any equilibrium. This paper will restudy this circuit, showing that two hidden attractors can coexist in this circuit for some parameters, and characterizes the basins of these two attractors by means of computer method as well. In addition, a computer-assisted proof of the chaoticity of these attracters is presented by a topological horseshoe theory.

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References

  1. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmospheric Sci. 20(2), 130–141 (1963)

    Article  Google Scholar 

  2. Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)

    Article  Google Scholar 

  3. Chua, L.O., Lin, G.-N.: Canonical realization of Chua’s circuit family. IEEE Trans. Circuits Syst. 37(7), 885–902 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bilotta, E., Pantan, P.: A gallery of Chua attractors. Nonlinear Sciences Series A. World Scientific Hackensack, Hackensack (2008)

    Google Scholar 

  5. Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurcation Chaos 9(07), 1465–1466 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Localization of hidden Chua’s attractors. Phys. Lett. A 375(23), 2230–2233 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Kuznetsov, N., Leonov, G., Vagaitsev, V.: Analytical-numerical method for attractor localization of generalized Chua’s system. IFAC Proc. Vol. (IFAC-PapersOnline) 4(1), 29–33 (2010)

    MathSciNet  Google Scholar 

  8. Kuznetsov, N., et al.: Analytical-numerical localization of hidden attractor in electrical Chua’s circuit. Informatics in Control, Automation and Robotics. Springer, Berlin (2013)

    Google Scholar 

  9. Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Hidden attractor in smooth Chua systems. Phys. D 241(18), 1482–1486 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bragin, V., et al.: Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits. J. Comput. Syst. Sci. Int. 50(4), 511–543 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Leonov, G.A., Kuznetsov, N.V.: Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int. J. Bifurcation Chaos 23(01), 1330002 (2013)

    Article  MathSciNet  Google Scholar 

  12. Li, Q., Zhou, H., Yang, X.-S.: A study of basin of attraction of the simplest walking model based on heterogeneous computation. Acta Phys. Sin 61(4), 040503 (2012)

    MathSciNet  Google Scholar 

  13. Li, Q., Yang, X.S.: New walking dynamics in the simplest passive bipedal walking model. Appl. Math. Model. 36(11), 5262–5271 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Lozi, R., Ushiki, S.: The theory of confinors in Chua’s circuit: accurate analysis of bifurcations and attractors. Int. J. Bifurcation Chaos 3(02), 333–361 (1993)

    Google Scholar 

  15. Leonov, G.A., Kuznetsov, N.V.: Time-varying linearization and the Perron effects. Int. J. Bifurcation Chaos 17(04), 1079–1107 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kuznetsov, N., Leonov, G.: Stability by the first approximation for discrete systems. VESTNIK-ST PETERSBURG UNIVERSITY MATHEMATICS C/C OF VESTNIK-SANKT-PETERBURGSKII UNIVERSITET MATEMATIKA. 36, 21–27 (2003)

    MATH  MathSciNet  Google Scholar 

  17. Chen, T., Choudhury, S.R.: Bifurcations and chaos in a modified driven chen’s system. Far E. J. Dyn. Sys. 18(1), 1–81 (2012)

    MathSciNet  Google Scholar 

  18. Gambino, G., Choudhury, S.R., Chen, T.: Modified post-bifurcation dynamics and routes to chaos from double-Hopf bifurcations in a hyperchaotic system. Nonlinear Dyn. 69(3), 1439–1455 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Li, Q., Tang, S., Yang, X.-S.: New bifurcations in the simplest passive walking model. Chaos: an interdisciplinary. J. Nonlinear Sci. 23(4), 043110 (2013)

    Google Scholar 

  20. Gritli, H., Khraief, N., Belghith, S.: Period-three route to chaos induced by a cyclic-fold bifurcation in passive dynamic walking of a compass-gait biped robot. Commun. Nonlinear Sci. Numer Simul. 17(11), 4356–4372 (2012)

    Article  MathSciNet  Google Scholar 

  21. Chen, Z.-M., Price, W.: Transition to chaos in a fluid motion system. Chaos, Solitons & Fractals 26(4), 1195–1202 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gambino, G., Lombardo, M.C., Sammartino, M.: Adaptive control of a seven mode truncation of the Kolmogorov flow with drag. Chaos, Solitons & Fractals 41(1), 47–59 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Boughaba, S., Lozi, R.: Fitting trapping regions for Chua’s attractor—a novel method based on isochronic lines. Int. J. Bifurcation Chaos 10(01), 205–225 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Yang, X.-S.: Topological horseshoes and computer assisted verification of chaotic dynamics. Int. J. Bifurcation Chaos 19(4), 1127–1145 (2009)

    Article  MATH  Google Scholar 

  25. Yang, X.S., Li, H., Huang, Y.: A planar topological horseshoe theory with applications to computer verifications of chaos. J. Phys. A 38(19), 4175–4185 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  26. Li, Q., Yang, X.-S.: A simple method for finding topological horseshoes. Int. J. Bifurcation Chaos 20(2), 467–478 (2010)

    Article  MATH  Google Scholar 

  27. Yang, X.-S., Tang, Y.: Horseshoes in piecewise continuous maps. Chaos, Solitons & Fractals 19(4), 841–845 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  28. Fan, Q.J.: Horseshoe chaos in a hybrid planar dynamical system. Int. J. Bifurcation Chaos 22(8), 1250202 (2012)

    Article  Google Scholar 

  29. Li, Q.D., Yang, X.S., Chen, S.: Hyperchaos in a spacecraft power system. Int. J. Bifurcation Chaos 21(6), 1719–1726 (2011)

    Article  MATH  Google Scholar 

  30. Li,Q., et al.: Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria and its implementation. Int. J. Circuit Theory Appl. (2013). doi:10.1002/cta.1912

  31. Li, Q.-D., SongTang, : Algorithm for finding Horseshoes in three-dimensional hyperchaotic maps and its application. Acta Phys. Sin 62(2), 020510 (2013)

    Google Scholar 

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Acknowledgments

This work is supported in part by National Natural Science Foundation of China (61104150), Science Fund for Distinguished Young Scholars of Chongqing (cstc2013jcyjjq40001), and the Science and Technology Project of Chongqing Education Commission (No. KJ130517).

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Authors and Affiliations

  1. Key Laboratory of Industrial Internet of Things & Networked Control, Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing , 400065, China

    Qingdu Li, Hongzheng Zeng & Xiao-Song Yang

  2. Department of Mathematics, Huazhong University of Science and Technology, Wuhan , 430074, China

    Xiao-Song Yang

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  1. Qingdu Li
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  2. Hongzheng Zeng
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  3. Xiao-Song Yang
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Correspondence to Xiao-Song Yang.

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Li, Q., Zeng, H. & Yang, XS. On hidden twin attractors and bifurcation in the Chua’s circuit. Nonlinear Dyn 77, 255–266 (2014). https://doi.org/10.1007/s11071-014-1290-8

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