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Estimations for ultimate boundary of a new hyperchaotic system and its simulation

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Abstract

This paper has investigated the boundedness of a new hyperchaotic Rabinovich system. We have obtained the global exponential attractive set and the ultimate bound Ω λ for this system. Furthermore, we can conclude that the rate of the trajectories of the system going from the exterior of the set Ω λ,2 to the interior of the set Ω λ,2 is an exponential rate. The estimate of the trajectories rate is also obtained. Numerical simulations are presented to show the effectiveness of the proposed scheme.

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References

  1. Liao, X., Fu, Y., Xie, S., Yu, P.: Globally exponentially attractive sets of the family of Lorenz systems. Sci. China, Ser. F 51, 283–292 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chen, G., Lü, J.: Dynamical Analysis, Control and Synchronization of the Lorenz Systems Family. Science Press, Beijing (2003)

    Google Scholar 

  3. Leonov, G., Bunin, A., Koksch, N.: Attractor localization of the Lorenz system. Z. Angew. Math. Mech. 67, 649–656 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  4. Leonov, G.: Lyapunov dimension formulas for Henon and Lorenz attractors. St. Petersburg Math. J. 13, 1–12 (2001)

    Google Scholar 

  5. Zhang, F., Shu, Y., Yang, H., Li, X.: Estimating the ultimate bound and positively invariant set for a synchronous motor and its application in chaos synchronization. Chaos Solitons Fractals 44, 137–144 (2011)

    Article  MATH  Google Scholar 

  6. Zhang, F., Shu, Y., Yang, H.: Bounds for a new chaotic system and its application in chaos synchronization. Commun. Nonlinear Sci. Numer. Simul. 16, 1501–1508 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Jian, J., Zhao, Z.: New estimations for ultimate boundary and synchronization control for a disk dynamo system. Nonlinear Anal. Hybrid Syst. 9, 56–66 (2013)

    Article  MathSciNet  Google Scholar 

  8. Zhang, F., Mu, C., Li, X.: On the boundedness of some solutions of the Lü system. Int. J. Bifurc. Chaos Appl. Sci. Eng. 22, 1250015 (2012)

    Article  MathSciNet  Google Scholar 

  9. Zhang, F., Li, Y., Mu, C.: Bounds of solutions of a kind of hyper-chaotic systems and application. J. Math. Res. Appl. 33(3), 345–352 (2013)

    MATH  MathSciNet  Google Scholar 

  10. Zhang, J., Tang, W.: A novel bounded 4D chaotic system. Nonlinear Dyn. 67, 2455–2465 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Pikovski, A., Rabinovich, M., Trakhtengerts, V.: Onset of stochasticity in decay confinement of parametric instability. Sov. Phys. JETP 47, 715–719 (1978)

    Google Scholar 

  12. Xie, F., Zhang, X.: Invariant algebraic surfaces of the Rabinovich system. J. Phys. A 36, 499–516 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Llibre, J., Messias, M., Silva, P.: On the global dynamics of the Rabinovich system. J. Phys. A 41, 1–21 (2008)

    Article  Google Scholar 

  14. Zhang, X.: Integrals of motion of the Rabinovich system. J. Phys. A 33, 5137–5155 (2003)

    Google Scholar 

  15. Liu, Y.: Hyperchaotic system from controlled Rabinovich system. IET Control Theory Appl. 28, 1671–1678 (2011)

    Google Scholar 

  16. Leonov, G., Kuznetsov, N.: Time-varying linearization and the Perron effects. Int. J. Bifurc. Chaos Appl. Sci. Eng. 17, 1079–1107 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Leonov, G., Kuznetsov, N.: Hidden attractors in dynamical systems. From hidden oscillations in Hilbert–Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits. Int. J. Bifurc. Chaos Appl. Sci. Eng. 23, 1330002 (2013)

    Article  MathSciNet  Google Scholar 

  18. Bragin, V., Vagaitsev, V., Kuznetsov, N., Leonov, G.: Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits. J. Comput. Syst. Sci. Int. 50, 511–543 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Boichenko, V., Leonov, G., Reitmann, V.: Dimension Theory for Ordinary Differential Equations. Teubner, Wiesbaden (2005)

    Book  MATH  Google Scholar 

  21. Leonov, G., Ponomarenko, D., Smirnova, V.: Frequency Domain Methods for Nonlinear Analysis: Theory and Applications. World Scientific, Singapore (1996)

    MATH  Google Scholar 

  22. Leonov, G., Boichenko, V.: Lyapunov’s direct method in the estimation of the Hausdorff dimension of attractors. Acta Appl. Math. 26, 1–60 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  23. Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos Appl. Sci. Eng. 9, 1465–1466 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurc. Chaos Appl. Sci. Eng. 12, 659–661 (2002)

    Article  MATH  Google Scholar 

  25. Leonov, G.: General existence conditions of homoclinic trajectories in dissipative systems. Lorenz, Shimizu–Morioka, Lu and Chen systems. Phys. Lett. A 376, 3045–3050 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  26. Lü, J., Chen, G.: Generating multi-scroll chaotic attractors: theories, methods and applications. Int. J. Bifurc. Chaos Appl. Sci. Eng. 16, 775–858 (2006)

    Article  MATH  Google Scholar 

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Acknowledgements

This work is supported in part by the Fundamental Research Funds for the Central Universities (No. CDJXS11100026) and NSF of China (No. 11371384). The authors are grateful to thank the anonymous referees and editors for their valuable comments and suggestions.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Fuchen Zhang, Chunlai Mu & Xianzhong Yao

  2. School of Mathematic and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing, 404000, China

    Liangwei Wang

  3. School of Electronic & Information Engineering, Dalian University of Technology, Dalian, 116024, China

    Xingyuan Wang

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  1. Fuchen Zhang
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  2. Chunlai Mu
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  3. Liangwei Wang
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Correspondence to Fuchen Zhang.

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Zhang, F., Mu, C., Wang, L. et al. Estimations for ultimate boundary of a new hyperchaotic system and its simulation. Nonlinear Dyn 75, 529–537 (2014). https://doi.org/10.1007/s11071-013-1082-6

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  • DOI: https://doi.org/10.1007/s11071-013-1082-6

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