An experimental verification of basin metamorphoses in a nonlinear mechanical system

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 7, Issue 6, 1997, Pages 1337-1357

  Todd, M D ^a,b   Virgin, L N ^a  

^a Duke University   (United States)

^b NAVAL RESEARCH LABORATORY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0039505985     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127497001060     Document Type: Article

References (36)

1. Albano, A. M., Muench, J., Schwartz, C., Mees, A. I. & Rapp, P. E. [1988] "Singular value decomposition and the Grassberger-Procaccia algorithm," Phys. Rev. A38, 3017-3026.
2. Broomhead, D. S. & King, G. P. [1986] "Extracting qualitative dynamics from experimental data," Physica D20, 217-236.
3. Cusumano, J. P. & Kimble, B. W. [1994] "Experimental verification of basins of attraction and homoclinic bifurcation in a magneto-mechanical oscillator," in Nonlinearity and Chaos in Engineering Dynamics, eds. Thompson, J. M. T. & Bishop, S. R. (Wiley and Sons, New York).
4. Cusumano, J. P. & Kimble, B. W. [1995] "A stochastic interrogation method for experimental measurements of global dynamics and basin evolution: Application to a two-well oscillator," Nonlin. Dynam. 8, 213-235.
5. Eschenazi, E., Solari, H. G. & Gilmore, R. [1989] "Basins of attraction in driven dynamical systems," Phys. Rev. A39, 2609-2627.
6. Gottwald, J. A., Virgin, L. N. & Dowell, E. H. [1992] "Experimental mimicry of Duffing's equation," J. Sound Vib. 158(3), 447-467.
7. Gottwald, J. A., Virgin, L. N. & Dowell, E. H. [1995] "Routes to escape from an energy well," J. Sound Vib. 187(1), 133-144.
8. Grebogi, C., Ott, E. & Yorke, J. A. [1987] "Basin boundary metamorphoses: Changes in accessible boundary orbits," Physica D24, 243-262.
9. Henon, M. [1982] "On the numerical computation of Poincaré maps," Physica 5D, 412-414.
10. Melnikov, V. K. [1963] "On the stability of the center for time periodic solutions," Trans. Moscow Math. Soc. 12, 1-57.
11. Nayfeh, A. H. & Balachandran, B. [1995] Applied Nonlinear Dynamics (Wiley and Sons, New York).
12. Nusse, H. E., Ott, E. & Yorke, J. A. [1995] "Saddle-node bifurcations on fractal basin boundaries," Phys. Rev. Lett. 75(13), 2482-2485.
13. Nusse, H. E. & Yorke, J. A. [1996] "Basins of attraction," Science 271, 1376-1380.
14. Nusse, H. E. & Yorke, J. A. [1997] "Wada basins and basin cells," Physica D90, 242-261.
15. Pezeshki, C. & Dowell, E. H. [1987] "An examination of initial condition maps for the sinusoidally excited buckled beam modeled by Duffing's equation," J. Sound Vib. 117(2), 291-232.
16. Shaw, S. W. & Haddow, A. G. [1992] "On 'roller-coaster' experiments for nonlinear oscillators," Nonlin. Dynam. 3, 375-384.
17. Soliman, M. S. [1993] "Jumps to resonance: Long chaotic transients, unpredictable outcome, and the probability of restabilization," J. Appl. Mech. 60(3), 669-676.
18. Soliman, M. S. [1995] "Dynamic bifurcations in non-stationary systems: Transitions with an unpredictable outcome," Proc. Roy. Soc. London A451, 471-485.
19. Soliman, M. S. [1996] "Jump phenomena resulting in unpredictable dynamics in the driven damped pendulum," Int. J. Nonlin. Mech. 31(2), 167-174.
20. Soliman, M. S. & Thompson, J. M. T. [1989] "Integrity measures quantifying the erosion of smooth and fractal basins of attraction," J. Sound Vib. 135(3), 453-475.
21. Soliman, M. S. & Thompson, J. M. T. [1990] "Stochastic penetration of smooth and fractal basin boundaries under noise excitation," Dynamics and Stability of Systems 5(4), 281-298.
22. Soliman, M. S. & Thompson, J. M. T. [1991] "Basin organization prior to a tangled saddle-node bifurcation," Int. J. Bifurcation and Chaos 1(1), 107-118.
23. Soliman, M. S. & Thompson, J. M. T. [1992a] "Global dynamics underlying sharp basin erosion in nonlinear driven oscillators," Phys. Rev. A45(6), 3425-3431.
24. Soliman, M. S. & Thompson, J. M. T. [1992b] "Indeterminate sub-critical bifurcations in parametric resonance," Proc. Roy. Soc. London A438, 511-518.
25. Soliman, M. S. & Thompson, J. M. T. [1992c] "Indeterminate trans-critical bifurcations in parametrically excited systems," Proc. Roy. Soc. London A439, 601-610.
26. Stewart, H. B. & Ueda, Y. [1991] "Catastrophes with indeterminate outcome," Proc. Roy. Soc. London A432, 113-123.
27. Thompson, J. M. T. [1989] "Chaotic phenomena triggering the escape from a potential well," Proc. Roy. Soc. London A421, 195-225.
28. Thompson, J. M. T., Bishop, S. R. & Leung, L. M. [1987] "Fractal basins and chaotic bifurcations prior to escape from a potential well," Phys. Lett. A121(3), 116-120.
29. Thompson, J. M. T. & MacRobie, F. A. [1993]. "Indeterminate bifurcations and the global dynamics of driven oscillators," in 1st European Nonlinear Oscillations Conf., eds. Kreuzer, E. & Schmidt, G., (Akademie Verlag, Berlin), pp. 107-128.
30. Thompson, J. M. T. & Soliman, M. S. [1990] "Fractal control boundaries of driven oscillators and their relevance to safe engineering design," Proc. Roy. Soc. London A428, 1-13.
31. Thompson, J. M. T. & Soliman, M. S. [1991] "Indeterminate jumps to resonance from a tangled saddle-node bifurcation," Proc. Roy. Soc. London A432, 101-111.
32. Todd, M. D. & Virgin, L. N. [1996] "An experimental impact oscillator," Chaos, Solitons, and Fractals, 8(4), 699-714.
33. Todd, M. D. & Virgin, L. N. [1997] "Natural frequency considerations of an impact oscillator," J. Sound Vib. 194(3), 452-460.
34. Todd, M. D., Virgin, L. N. & Gottwald, J. A. [1996] "The nonstationary transition through resonance," Nonlin. Dynam. 10, 31-48.
35. Tongue, B. H. [1987] "On obtaining global nonlinear system characteristics through interpolated cell mapping," Physica D28, 401-408.
36. Ueda, Y., Yoshida, S., Stewart, H. B. & Thompson, J. M. T. [1990] "Basin explosions and escape phenomena in the twin-well duffing oscillator: Compound global bifurcations organizing behavior," Phil. Trans. Roy. Soc. London 332, 169-186.