Effects of Multi Global Bifurcations on Basin Organization, Catastrophes and Final Outcomes in a Driven Nonlinear Oscillator at the 2T-Subharmonic Resonance

Nonlinear Dynamics

Volumn 17, Issue 1, 1998, Pages 41-59

  Szemplińska Stupnicka, Wanda ^a   Tyrkiel, Elzbieta ^a  

^a INSTITUTE OF FUNDAMENTAL TECHNOLOGICAL RESEARCH   (Poland)

Author keywords

2T subharmonic resonance; Escape oscillator; Fractal basin boundaries; Global bifurcations; Unpredictable outcomes

Indexed keywords

BIFURCATION (MATHEMATICS); FRACTALS; NUMERICAL ANALYSIS; RESONANCE;

ESCAPE OSCILLATORS;

OSCILLATORS (ELECTRONIC);

EID: 0032156602     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1008299800048     Document Type: Article

References (23)

1. Soliman, M. S. and Thompson, J. M. T., 'Basin organization prior to a tangled saddle-node bifurcation', International Journal of Bifurcation and Chaos 1(1), 1991, 107-118.
2. Soliman, M. S. and Thompson, J. M. T., 'The effect of damping on the steady state and basin bifurcation patterns of a nonlinear mechanical oscillator', International Journal of Bifurcation and Chaos 2(1), 1992, 81-91.
3. Thompson, J. M. T. and Soliman, M. S., 'Indeterminate jumps to resonance from a tangled saddle-node bifurcation', Proceedings of the Royal Society of London A432, 1991, 101-111.
4. Soliman, M. S., 'Jumps to resonance: Long chaotic transients, unpredictable outcome, and the probability of restabilization', Journal of Applied Mechanics 60, 1993, 669-676.
5. Grebogi, C., Ott, E., and Yorke, J. A., 'Crises, sudden changes in chaotic attractors and transient chaos', Physica D7, 1983, 181-200.
6. McDonald, S. W., Grebogi, C., Ott, E., and Yorke, J. A., 'Fractal basin boundaries', Physica D17, 1985, 125-153.
7. Wiggins S., Global Bifurcations and Chaos: Analytical Methods, Springer-Verlag, New York, 1988.
8. Ott, E., Chaos in Dynamical Systems, Cambridge University Press, Cambridge, 1993.
9. Soliman, M. S. and Thompson, J. M. T., 'Indeterminate sub-critical bifurcations in parametric resonance', Proceedings of the Royal Society of London A438, 1992, 511-518.
10. Soliman, M. S. and Thompson, J. M. T., 'Indeterminate trans-critical bifurcations in parametrically excited systems', Proceedings of the Royal Society of London A439, 1992, 601-610.
11. Soliman, M. S., 'Predicting regimes of indeterminate jumps to resonance by assessing fractal boundaries in control space', International Journal of Bifurcation and Chaos 4(6), 1994, 1645-1653.
12. Soliman, M. S., 'Dynamic bifurcations in nonstationary systems: Transitions with an unpredictable outcome', Proceedings of the Royal Society of London A451, 1995, 471-485.
13. Soliman, M. S., 'Sub-critical flip bifurcations that may or may not result in period-doubling', International Journal of Bifurcation and Chaos 6(4), 1996, 751-757.
14. Stewart, H. B. and Ueda, Y., 'Catastrophes with indeterminate outcome', Proceedings of the Royal Society of London A432, 1991, 113-123.
15. Szemplińska-Stupnicka, W. and Tyrkiel, E., 'Sequences of global bifurcations and the related outcomes after crisis of the resonant attractor in a nonlinear oscillator', International Journal of Bifurcation and Chaos 7(11), 1997, 2437-2457.
16. Stoker, J. I., Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience, New York, 1950.
17. Hayashi, Ch., Nonlinear Oscillations in Physical Systems, Princeton University Press, Princeton, NJ, 1985.
18. Szemplińska-Stupnicka, W., 'Secondary resonances and approximate models of routes to chaotic motions in nonlinear oscillators', Journal of Sound and Vibration 113, 1987, 155-172.
19. Szemplińska-Stupnicka, W., The Behavior of Nonlinear Vibrating Systems, Vol. I, Fundamental Concepts and Methods: Applications to Single-Degree-of-Freedom Systems, Kluwer Academic Publishers, Dordrecht, 1990.
20. Szemplińska-Stupnicka, W., The Behavior of Nonlinear Vibrating Systems, Vol. II, Advanced Concepts and Applications to Multi-Degree-of-Freedom Systems, Kluwer Academic Publishers, Dordrecht, 1990.
21. Janicki, K. L. and Szemplińska-Stupnicka, W., 'Subharmonic resonances and criteria for escape and chaos in a driven oscillator', Journal of Sound and Vibration 180, 1995, 253-269.
22. Janicki, K. L. and Szemplińska-Stupnicka, W., 'Subharmonic resonances in a driven oscillator: Bifurcation structures and transitions to chaos', European Journal of Mechanics and Solids 16(4), 1997, 671-694.
23. Nusse, H. E. and Yorke, J. A., Dynamics: Numerical Explorations, Springer-Verlag, New York, 1994.