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Chaos control in a pendulum system with excitations and phase shift

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Abstract

Melnikov methods are used for suppressing homoclinic and heteroclinic chaos of a pendulum system with a phase shift and excitations. This method is based on the addition of adjustable amplitude and phase-difference of parametric excitation. Theoretically, we give the criteria of suppression of homoclinic and heteroclinic chaos, respectively. Numerical simulations are given to illustrate the effect of the chaos control in this system. Moreover, we calculate the maximum Lyapunov exponents (LEs) in parameter plane, and study how to vary the maximum LE when the parametric frequency varies.

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References

  1. Alasty, A., Salarieh, H.: Nonlinear feedback control of chaotic pendulum in presence of saturation effect. Chaos Solitons Fractals 31(2), 292–304 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baker, G.L.: Control of the chaotic driven pendulum. Am. J. Phys. 63(9), 832–838 (1995)

    Article  Google Scholar 

  3. Cao, H.J., Chi, X.B., Chen, G.R.: Suppressing or inducing chaos by weak resonant excitations in an externally-forced froude pendulum. Int. J. Bifurc. Chaos 14(3), 1115–1120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cao, H.J., Chen, G.R.: Global and local control of homoclinic and heteroclinic bifurcation. Int. J. Bifurc. Chaos 15(8), 2411–2432 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chacón, R.: Natural symmetries and regularization by means of weak parametric modulations in the forced pendulum. Phys. Rev. E 52, 2330–2337 (1995)

    Article  MathSciNet  Google Scholar 

  6. Chacón, R.: General results on chaos suppression for biharmonically driven dissipative systems. Phys. Lett. A 257, 293–300 (1999)

    Article  Google Scholar 

  7. Chacón, R., Palmero, F., Balibrea, F.: Taming chaos in a driven Josephson junction. Int. J. Bifurc. Chaos 11(7), 1897–1909 (2001)

    Article  Google Scholar 

  8. Chacón, R.: Relative effectiveness of weak periodic excitations in suppressing homoclinic heteroclinic chaos. Eur. Phys. J. B 65, 207–210 (2002)

    Article  Google Scholar 

  9. Chen, G.R., Dong, X.: From Chaos to Order: Methodologies, Perspectives and Applications. World Scientific, Singapore (1998)

  10. Chen, G.R., Moiola, J., Wang, H.O.: Bifurcation control: theories, methods and applications. Int. J. Bifurc. Chaos 10(3), 511–548 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, X., Fu, X., Jing, Z.: Complex dynamics in a pendulum equation with a phase shift. Int. J. Bifurc. 22(12), 387–426 (2012)

    MathSciNet  Google Scholar 

  12. D’Humieres, D., Beasley, M.R., Huberman, B.A., Libchaber, A.F.: Chaotic states and routes to chaos in the forced pendulum. Phys. Rev. A 26(6), 3483–3492 (1982)

    Article  Google Scholar 

  13. Jing, Z.J., Yang, J.P.: Complex dynamics in pendulum equation with parametric and external excitations (I). Int. J. Bifurc. Chaos 10(16), 2887–2902 (2006)

    Article  MathSciNet  Google Scholar 

  14. Jing, Z.J., Yang, J.P.: Complex dynamics in pendulum equation with parametric and external excitations (II). Int. J. Bifurc. Chaos 10(16), 3053–3078 (2006)

    Article  MathSciNet  Google Scholar 

  15. Kapitaniak, T.: Introduction. Chaos Solitons Fractals 15, 201–203 (2003)

  16. Lenci, S., Rega, G.: A procedure for reducing the chaotic response region in an impact mechanical system. Nonlinear Dyn. 15, 391–409 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lenci, S., Rega, G.: Optimal control of nonregular dynamics in a Duffing oscillator. Nonlinear Dyn. 33, 71–86 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lenci, S., Rega, G.: Optimal control of homoclinic bifurcation: theoretical treatment and practical reduction of safe basinerosion in the Helmholtz oscillator. J. Vib. Control 9(3), 281–316 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)

    MATH  Google Scholar 

  20. Ott, E., Grebogi, N., Yorke, J.: Controlling chaos. Phys. Rev. Lett. 64(11), 1196–1199 (1990)

  21. Perira-Pinto, F.H.I., Ferreira, A.M., Savi, M.A.: Chaos control in a nonlinear pendulum using a semi-continuous method. Chaos Solitons Fractals 22, 653–668 (2004)

    Article  Google Scholar 

  22. Shinbrot, T., Ott, E., Grebogi, N., Yorke, J.: Using chaos to direct trajectories to targets. Phys. Rev. Lett. 65, 3215–3218 (1990)

    Article  Google Scholar 

  23. Wang, R.Q., Jing, Z.J.: Chaos control of chaotic pendulum system. Chaos Solitons Fractals 21, 201–207 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wiggins, S.: Global bifurcation and chaos: analytical methods. Springer, Berlin (1988)

    Book  Google Scholar 

  25. Yagasaki, K., Uozumi, T.: Controlling chaos in a pendulum subjected to feedforward and feedback control. Int. J. Bifurc. Chaos 7(12), 2827–2835 (1997)

    Article  MATH  Google Scholar 

  26. Yagasaki, K.: Dynamics a pendulum subjected to feedforward and feedback control. JSME. Int. J. 41(3), 545–554 (1998)

    Article  Google Scholar 

  27. Yang, J.P., Jing, Z.J.: Inhibition of chaos in a pendulum equation. Chaos Solitons Fractals 35, 726–737 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors would like to thank the reviewers for their helpful comments and suggestions. This work is supported by the National Natural Science Foundation of China (No. 11071066 and No. 11171206).

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan , 411201, People’s Republic of China

    Xianwei Chen & Xiangling Fu

  2. College of Mathematics and Computer Science , Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), Hunan Normal University, Changsha , 410081, Hunan, People’s Republic of China

    Xianwei Chen & Zhujun Jing

  3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing , 100190, People’s Republic of China

    Zhujun Jing

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  1. Xianwei Chen
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  2. Zhujun Jing
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Correspondence to Xianwei Chen.

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Chen, X., Jing, Z. & Fu, X. Chaos control in a pendulum system with excitations and phase shift. Nonlinear Dyn 78, 317–327 (2014). https://doi.org/10.1007/s11071-014-1441-y

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  • DOI: https://doi.org/10.1007/s11071-014-1441-y

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