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On the stability "in the large'' and unsafe disturbances in a nonlinear oscillator

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The problem of the stability "in the large'' and the unsafe disturbances of the equilibrium position is studied for the structures whose dynamics is governed by the equation of motion of the pendulum with parametric excitation. The system displays a variety of nonlinear and chaotic phenomena, so that the study requires the use of theoretical concepts of the mathematics of chaos. Detailed explorations are performed by the aid of the nonlinear software package Dynamics.
Rocznik
Strony
155--168
Opis fizyczny
Bibliogr. 31 poz., wykr.
Twórcy
  • Institute of Fundamental Technological Research, Polish Academy of Sciences ul. Świętokrzyska 21, 00-049 Warsaw, Poland
autor
  • Institute of Fundamental Technological Research, Polish Academy of Sciences ul. Świętokrzyska 21, 00-049 Warsaw, Poland
autor
  • Institute of Fundamental Technological Research, Polish Academy of Sciences ul. Świętokrzyska 21, 00-049 Warsaw, Poland
Bibliografia
  • [1] S.R. Bishop, M.J. Clifford. Non rotating periodic orbits in the parametrically excited pendulum. Eur. J. Mech. A/Solids, 17: 581-587, 1994.
  • [2] S.R. Bishop, M.J. Clifford. Zones of chaotic behavior in the parametrically excited pendulum. J. Sound and Vibration, 189(1): 142-147, 1996.
  • [3] D. Capecchi. Geometric aspects of the parametrically driven pendulum. Nonlinear Dynamics, 7: 231-247, 1995.
  • [4] D. Capecchi, S.R. Bishop. Periodic oscillations and attracting basins for a parametrically excited pendulum. Dynamics and Stability of Systems, 9(2): 123-143, 1994.
  • [5] M.J. Clifford, S.R. Bishop. Bifurcational precedences for parametric escape from a symmetric potential well. Int. J. Bifurcation and Chaos, 4(3): 623-630, 1994.
  • [6] C. Grebogi, E. Ott, J.A. Yorke. Crises, sudden changes in chaotic attractors and transient chaos. Physica, D7: 181-200, 1983.
  • [7] J. Guckenheimer, P.J. Holmes. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York, 1983.
  • [8] Ch. Hayashi. Nonlinear Oscillations in Physical Systems. Princeton University Press, Princeton, N.J., 1985.
  • [9] G.W. Housner. The behavior of inverted pendulum structure during earthquake. Bul. Seismological Soc. America, 53: 403-417, 1963.
  • [10] S. Kaliski. Vibrations and Waves. PWN, Warsaw, Polish edition 1964, English edition 1985.
  • [11] B.P Koch, R.W. Leven. Subharmonic and homoclinic bifurcations in a parametrically forced pendulum. Physica D16: 1-13, 1985.
  • [12] S.W. McDonald, C. Grebogi, E. Ott, J.A. Yorke. Fractal basin boundaries. Physica, D17: 125-153, 1985.
  • [13] N. Minorski. Nonlinear Oscillations. Nostrand Comp., New York, 1962.
  • [14] H.E. Nusse, J.A. Yorke. Dynamics: Numerical Explorations. Springer-Verlag, New York, 1998.
  • [15] E. Ott. Chaos in Dynamical Systems. Cambridge University Press, Cambridge, 1993.
  • [16] R.C.T. Rainey. The dynamics of tethered platform. Trans. Roy. Inst. Naval Architects, 420: 59-80, 1978.
  • [17] F.M. Salam, S.S. Sastry. Dynamics of the forced Josephson junction — the region of chaos. IEEE Trans. Circuits and Systems CAS, 30: 784-796, 1985.
  • [18] M.S. Soliman. Predicting regimes of indeterminate jumps to resonance by assessing fractal boundaries in control space. Int. J. Bifurcation and Chaos, 4(6): 1645-1653, 1994.
  • [19] M.S. Soliman, J.M.T. Thompson. Integrity measures quantifying the erosion of smooth and fractal basins of attraction. J. Sound and Vibration, 135(3): 453-467, 1989.
  • [20] W. Szemplińska-Stupnicka. The Behavior of Nonlinear Vibrating Systems; vol. I - Fundamental Concepts and Methods: Applications to Single-Degree-of-Freedom Systems. Kluwer Academic Publishers, Dordrecht-Boston-London, 1990.
  • [21] W. Szemplińska-Stupnicka, E. Tyrkiel. Sequences of global bifurcations and the related outcomes after crisis of the resonant attractor in a nonlinear oscillator. Int. J. Bifurcation and Chaos, 7(11): 2437-2457, 1997.
  • [22] W. Szemplińska-Stupnicka, A. Zubrzycki, E. Tyrkiel. Properties of chaotic and regular boundary crisis indissipative driven nonlinear oscillators. Nonlinear Dynamics, 19: 19-36, 1999.
  • [23] W. Szemplińska-Stupnicka, E. Tyrkiel. Effects of multi global bifurcations on basin organization, catastrophes and final outcomes in a driven nonlinear oscillator at the 2T-subharmonic resonance. Nonlinear Dynamics, 17: 41-59, 1998.
  • [24] J.M.T. Thompson. Chaotic phenomena triggering the escape from the potential well. Proc. Roy. Soc. Lond., A421: 195-225, 1989. 168 W. Szemplińska-Stupnicka, E. Tyrkiel and A. Zubrzycki Computer Assisted Mechanics and Engineering Sciences, 8: 169-182,
  • [25] J.M.T. Thompson. Loss of engineering integrity due to erosion of absolute and transient basin boundaries. In: W. Schiehlen, ed., Nonlinear Dynamics in Engineering Systems, 313-320. Springer-Verlag, Berlin, 1990.
  • [26] J.M.T. Thompson, H.B. Stewart. Nonlinear Dynamics and Chaos. John Wiley and Sons, Chichester, 1986.
  • [27] J.M.T. Thompson, M.S. Soliman. Fractal control boundaries of driven oscillators and their relevance to safe engineering design. Proc. Roy. Soc. Lond., A428: 1-13, 1990.
  • [28] J.M.T. Thompson, H.B. Stewart, Y. Ueda. Safe, explosive and dangerous bifurcations in dissipative dynamical systems. Phys. Rev., E 49(2): 1019-1027, 1994.
  • [29] A. Tondl. A method of solving stability "in the large" with the aid of analog computers. Acta Technica CSAV, 5: 576-597, 1966.
  • [30] A. Tondl. Domains of attraction for nonlinear systems. Monographs and Memoranda, Bechovice, 11, 1970.
  • [31] S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York, 1990.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-BPB1-0006-0036
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