Modeling chaotic motions of a string from experimental data. (English) Zbl 0899.73008
Summary: Experimental measurements of nonlinear vibrations of a string are analyzed using new techniques of nonlinear modeling. Previous theoretical and numerical work suggested that the motions of a string can be chaotic and a Shil’nikov mechanism is responsible. We show that the experimental data is consistent with a Shil’nikov mechanism. We also reveal a period doubling cascade with a period three window which is not immediately observable because there is sufficient noise, probably of a dynamical origin, to mask the period-doubling bifurcation and the period three window.
MSC:
| 74-05 | Experimental work for problems pertaining to mechanics of deformable solids |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
Keywords:
nonlinear modeling; Shil’nikov mechanism; period doubling cascade; period-doubling bifurcation; period three windowReferences:
| [1] | Abarbanel, H. D.I.; Brown, R.; Sidorowich, J. J.; Tsimring, L. S., Analysis of Observed Chaotic Data, Vol. 65, 4 (1993) |
| [2] | Abarbanel, H. D.I.; Kennel, M. B., Local false nearest neighbors and dynamical dimensions from observed chaotic data, (Technical report (1992), Department of Physics, University of California: Department of Physics, University of California San Diego) · Zbl 0802.58041 |
| [3] | Akaike, H., A new look at the statistical identification model, IEEE Trans. Ann. Control, 19, 716-723 (1974) · Zbl 0314.62039 |
| [4] | Bajaj, A. K.; Johnson, J. M., On the amplitude dynamics and crisis in resonant motion of stretched strings, Phil. Trans. R. Soc. Lond. A, 338, 1-41 (1992) · Zbl 0753.58018 |
| [5] | Brown, R., Orthonormal polynomials as prediction functions in arbitrary phase space dimensions, (Technical report (1992), Institute for Nonlinear Science, University of California at San Diego) |
| [6] | Farmer, J. D.; Sidorowich, J. J., Optimal shadowing and noise reduction, Physica D, 47, 373-392 (1991) · Zbl 0729.65501 |
| [7] | Glendinning, P.; Sparrow, C., Bifurcations near homoclinic orbits, (XVIth International Congress of Theoretical and Applied Mechanics (1984)) · Zbl 0588.58041 |
| [8] | Glendinning, P.; Sparrow, C. T., Local and global behavior near homoclinic orbits, J. Stat. Phys., 35, 645-697 (1983) · Zbl 0588.58041 |
| [9] | Judd, K.; Mees, A. I., A model selection algorithm for nonlinear time series, Physica D, 82, 426-444 (1995) · Zbl 0888.58034 |
| [10] | Judd, K.; Mees, A. I., Modeling chaotic motions of a string from experimental data (1995), submitted for publication |
| [11] | Judd, K.; Mees, A. I., A simulated annealing approach model selection for highly nonlinear systems (1995), in preparation |
| [12] | Judd, K.; Mees, A. I.; Aihara, K.; Toyoda, M., Grid imaging for a two-dimensional map, Intern. J. Bifurc. & Chaos, 1, 1, 197-210 (1991) · Zbl 0755.92002 |
| [13] | Mees, A. I., Parsimonious dynamical reconstruction, International Journal of Bifurcation and Chaos, 3, 669-675 (1993) · Zbl 0875.62426 |
| [14] | Mees, A. I.; Sparrow, C. T., Some tools for analyzing chaos, (Proceedings IEEE, 75 (1987)), 1058-1070 |
| [15] | Molteno, T.; Tufillaro, N. B., J. Sound Vib., 137, 327 (1990) |
| [16] | Rissanen, J., (Stochastic Complexity in Statistical Inquiry, Vol. 15 (1989), World Scientific: World Scientific Singapore) · Zbl 0800.68508 |
| [17] | Sauer, T.; Yorke, J. A.; Embedology, M. Casdagli, J. Stat. Phys., 65, 579-616 (1992) · Zbl 0943.37506 |
| [18] | Schwarz, G., Estimating the dimension of a model, Ann. Stat., 6, 2, 461-464 (1977) · Zbl 0379.62005 |
| [19] | Shil’nikov, L. P., A case of the existence of a denumerable set of periodic motions, Sov. Math. Dokl., 6, 163-166 (1965) · Zbl 0136.08202 |
| [20] | Shil’nikov, L. P., On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle-type, Math. USSR Sb., 6, 427-438 (1968) · Zbl 0188.15303 |
| [21] | Takens, F., Detecting strange attractors in turbulence, (Rand, D. A.; Young, L. S., Dynamical Systems and Turbulence, Vol. 898 (1981), Springer: Springer Berlin), 365-381 · Zbl 0513.58032 |
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