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Deterministic chaos of a spherical pendulum under limited excitation

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Abstract

We investigate the appearance, development, and vanishing of deterministic chaos in a “spherical pendulum-electric motor of limited power” dynamical system. Chaotic attractors discovered in the system are described in detail.

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References

  1. N. N. Bogolyubov, “Perturbation theory in nonlinear mechanics,” Sb. Stroit. Mekh. Akad. Nauk Ukr. SSR, No. 14, 9–34 (1950).

  2. A. Ya. Gadionenko, “On periodic motions of a pendulum with vibrating point of suspension,” Ukr. Mat. Zh., 18, No. 5, 97–101 (1966).

    MATH  MathSciNet  Google Scholar 

  3. P. L. Kapitsa, “Dynamical instability of a pendulum with oscillating point of suspension,” Zh. Éksper. Teor. Fiz., 21, 588–607 (1951).

    Google Scholar 

  4. Yu. A. Mitropol’skii and A. Yu. Shvets, “On the influence of delay on the stability of a pendulum with vibrating point of suspension,” in: Analytic Methods for Investigation of Nonlinear Oscillations [in Russian], Institute of Mathematics, Academy of Sciences of Ukrainian SSR, Kiev (1980), pp. 115–120.

    Google Scholar 

  5. A. Yu. Shvets, “Influence of a varying delay on the stability of oscillations of a pendulum with vibrating suspension,” Ukr. Mat. Zh., 37, No. 1, 127–129 (1985).

    Article  MathSciNet  Google Scholar 

  6. I. A. Lukovskii, “Mechanical analogies in problems of nonlinear dynamics of a solid with liquid,” in: Problems of Mechanics and Its Applications, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (2002), pp. 125–160.

    Google Scholar 

  7. J. D. Crawford and E. Knobloch, “Symmetry and symmetry-breaking bifurcations in fluid dynamics,” Ann. Rev. Fluid Mech., 23, 341–387 (1991).

    MathSciNet  Google Scholar 

  8. T. Kambe and M. Umeki, “Nonlinear dynamics of two-mode interaction in parametric excitation of surface waves,” J. Fluid Mech., 139, 461–471 (1990).

    MathSciNet  Google Scholar 

  9. T. S. Krasnopolskaya and A. Yu. Shvets, “Chaotic surface waves in limited power-supply cylindrical tank vibrations,” J. Fluids Structures, 8, No. 1, 1–18 (1994).

    Article  Google Scholar 

  10. J. W. Miles, “Resonant motion of a spherical pendulum,” Physica D, 11, No. 3, 309–323 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  11. J. W. Miles, “Nonlinear Faraday resonance,” J. Fluid Mech., 146, No. 2, 285–302 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  12. V. S. Anishchenko, Complicated Oscillations in Simple Systems [in Russian], Nauka, Moscow (1990).

    Google Scholar 

  13. A. Yu. Shvets, “A chart of dynamical modes of a physical pendulum under limited excitation,” Zb. Prats’ Inst. Mat. Nats. Akad. Nauk Ukr., 1, No. 2, 197–209 (2004).

    Google Scholar 

  14. V. O. Kononenko, Vibrating System with a Limited Power-Supply, Iliffe, London (1969).

    Google Scholar 

  15. T. S. Krasnopolskaya and A. Yu. Shvets, “Chaos in vibrating systems with limited power-supply,” Chaos, 3, No. 3, 387–395 (1993).

    Article  Google Scholar 

  16. J. W. Miles, “Stability of forced oscillations of a spherical pendulum,” Quart. Appl. Math., 20, No. 1, 21–32 (1962).

    MATH  MathSciNet  Google Scholar 

  17. T. S. Krasnopolskaya and A. Yu. Shvets, “Chaotic oscillations of a spherical pendulum as the effect of interaction with excitation device,” in: Complexity in Physics and Technology, World Scientific, Singapore (1992), pp. 77–89.

    Google Scholar 

  18. N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1974).

    MATH  Google Scholar 

  19. Yu. A. Mitropol’skii, Averaging Method in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1971).

    Google Scholar 

  20. E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations. I. Nonstiff Problems, Springer, Berlin (1987).

    MATH  Google Scholar 

  21. G. Benettin, L. Galgani, and J. M. Strelcyn, “Kolmogorov entropy and numerical experiments,” Phys. Rev. A, 14, No. 6, 2338–2342 (1976).

    Article  Google Scholar 

  22. S. P. Kuznetsov, Dynamical Chaos [in Russian], Fizmatlit, Moscow (2001).

    Google Scholar 

  23. M. J. Feigenbaum, “Quantitative universality for a class of nonlinear transformations,” J. Statist. Phys., 19, No. 1, 25–52 (1978).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Kiev Polytechnic Institute, Kiev

    A. Yu. Shvets

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  1. A. Yu. Shvets
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__________

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 534–548, April, 2007.

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Shvets, A.Y. Deterministic chaos of a spherical pendulum under limited excitation. Ukr Math J 59, 602–614 (2007). https://doi.org/10.1007/s11253-007-0039-7

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  • DOI: https://doi.org/10.1007/s11253-007-0039-7

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