The global bifurcations that lead to transient tumbling chaos in a parametrically driven pendulum

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 10, Issue 9, 2000, Pages 2161-2175

  Szemplinska Stupnicka W ^a   Tyrkiel, E ^a   Zubrzycki, A ^a  

^a INSTITUTE OF FUNDAMENTAL TECHNOLOGICAL RESEARCH   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034257472     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127400001365     Document Type: Article

References (30)

1. Bishop, S. R. & Clifford, M. J. [1994] "Non rotating periodic orbits in the parametrically excited pendulum," Eur. J. Mech. A/Solids 17, 581-587.
2. Bishop, S. R. & Clifford, M. J. [1996] "Zones of chaotic behavior in the parametrically excited pendulum," J. Sound Vibr. 189(1), 142-147.
3. Bolotin, V. V. [1964] The Dynamic Stability of Elastic Systems (Holden-Day, San Francisco).
4. Capecchi, D. & Bishop, S. R. [1994] "Periodic oscillations and attracting basins for a parametrically excited pendulum," Dyn. Stab. Syst. 9(2), 123-143.
5. Capecchi, D. [1995] "Geometric aspects of the paramet rically driven pendulum," Nonlin. Dyn. 7, 231-247.
6. Clifford, M. J. & Bishop, S. R. [1994] "Bifurcational precedences for parametric escape from a symmetric potential well," Int. J. Bifurcation and Chaos 4(3), 623-630.
7. Floquet, G. [1883] "Sur les équations différentielles linéaires à coefficients périodiques," Ann. Ecole Norm. Super. 2(12), 47-89.
8. Grebogi, C., Ott, E. & Yorke, J. A. [1983] "Crises, sudden changes in chaotic attractors and transient chaos," Physica D7, 181-200.
9. Grebogi, C., Ott, E. & Yorke, J. A. [1987] "Basin boundary metamorphoses: Changes in accessible boundary orbits," Physica D24, 243-262.
10. Guckenheimer, J. & Holmes, P. J. [1983] Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, NY).
11. Hayashi, Ch. [1985] Nonlinear Oscillations in Physical Systems (Princeton University Press, Princeton, NJ).
12. Koch, B. P. & Leven, R. W. [1985] "Subharmonic and homoclinic bifurcations in a parametrically forced pendulum," Physica D16, 1-13.
13. McDonald, S. W., Grebogi, C., Ott, E. & Yorke, J. A. [1985] "Fractal basin boundaries," Physica D17, 125-153.
14. Nayfeh, A. H. [1993] Method of Normal Forms (Interscience, NY).
15. Nusse, H. E. & Yorke, J. A. [1998] Dynamics: Numerical Explorations (Springer-Verlag, NY).
16. Ott, E. [1993] Chaos in Dynamical Systems (Cambridge University Press, Cambridge).
17. Smale, S. [1967] "Differential dynamical system," Bull. Am. Math. Soc. 73, 747-817.
18. Soliman, M. S. & Thompson, J. M. T. [1992] "Indeterminate sub-critical bifurcations in parametric resonance," Proc. R. Soc. London A438, 511-518.
19. Soliman, M. S. [1994] "Predicting regimes of indeterminate jumps to resonance by assessing fractal bound aries in control space," Int. J. Bifurcation and Chaos 4(6), 1645-1653.
20. Soliman, M. S. [1995] "Dynamic bifurcations in non-stationary systems: Transitions with an unpredictable outcome," Proc. R. Soc. London A451, 471-485.
21. Szemplińska-Stupnicka, W. [1990] The Behavior of Nonlinear Vibrating Systems; Vol. II, Advanced Concepts and Applications to Multi-Degree-of-Freedom Systems (Kluwer Academic Publishers, Dordrecht/Boston/London).
22. Szemplińska-Stupnicka, W. & Tyrkiel, E. [1997] "Sequences of global bifurcations and the related outcomes after crisis of the resonant attractor in a non-linear oscillator," Int. J. Bifurcation and Chaos 7(11), 2437-2457.
23. Szemplińska-Stupnicka, W., Zubrzycki, A. & Tyrkiel, E. [1999] "Properties of chaotic and regular boundary crisis in dissipative driven nonlinear oscillators," Nonlin. Dyn. 19, 19-36.
24. Thompson, J. M. T. & Stewart, H. B. [1986] Nonlinear Dynamics and Chaos (John Wiley, Chichester).
25. Thompson, J. M. T., Stewart, H. B. & Ueda, Y. [1994] "Safe, explosive and dangerous bifurcations in dissipative dynamical systems," Phys. Rev E49(2), 1019-1027.
26. Wiggins, S. [1988] Global Bifurcations and Chaos: Analytical Methods (Springer-Verlag, NY).
27. Wiggins, S. [1990] Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, NY).
28. Wiggins, S. [1992] Chaotic Transport in Dynamical Systems (Springer-Verlag, NY).
29. Zavodney, L. D. & Nayfeh, A. H. [1988] "The response of a single degree of freedom system with quadratic and cubic nonlinearity to a fundamental parametric resonance," J. Sound Vibr. 120, 63-93.
30. Zavodney, L. D., Nayfeh, A. H. & Sanchez, N. E. [1990] "Bifurcations and chaos in parametrically excited single degree of freedom system," Nonlin. Dyn. 1, 1-21.