The characterization of a chaotic attractor in a driven, Duffing-Holmes oscillator with power-law damping is considered. State space reconstruction of the time series of the attractor is carried out to investigate its structure. The invariants associated with the attractor such as correlation dimension and entropy are computed. Also the maximum-likelihood (ML) estimation of dimension and entropy are carried out. The use of obtained invariants in building models for prediction and control using power-law dampers is discussed.
Issue Section:
Technical Papers
1.
Abarbanel
H. D. I.
1994
, “Predicting Physical Variables in Time-Delay Embedding
,” Physical Review E
, Vol. 49
, pp. 1840
–1853
.2.
Abhyankar
N. S.
Hall
E. K.
Hanagud
S. V.
1993
, “Chaotic Vibrations of Beams: Numerical Solution of Partial Differential Equations
,” ASME JOURNAL OF APPLIED MECHANICS
, Vol. 60
, pp. 167
–172
.3.
Badii
R.
1988
, “Dimension Increase in Filtered Chaotic Signals
,” Physical Review Letters
, Vol. 60
, pp. 979
–982
.4.
Braun, S. G., 1986, Mechanical Signature Analysis, Academic Press, London.
5.
Casdagli, M., and Eubank, S., 1991, Nonlinear Prediction and Modelling, Addison-Wesley, Reading, MA.
6.
Chennaoui
A.
1990
, “Attractor Reconstruction From Filtered Chaotic Time Series
,” Physical Review A
, Vol. 41
, pp. 4151
–4159
.7.
Eckmann
J.-P.
Ruelle
D.
1985
, “Ergodic Theory of Chaos and Strange Attractors
,” Reviews of Modern Physics
, Vol. 57
, pp. 617
–656
.8.
Farmer, J. D., and Sidorowich, J. J., 1987, Physical Review Letters, Vol. 59, pp. 845–848.
9.
Fraser
A. M.
1989
, “Information and Entropy in Strange Attractors
,” IEEE transactions on Information theory
, Vol. 35
, pp. 245
–262
.10.
Gersch
W.
Liu
R. S. Z.
1976
, “Time Series Methods for the Synthesis of Random Vibration Systems
,” ASME JOURNAL OF APPLIED MECHANICS
, Vol. 43
, pp. 159
–165
.11.
Gershenfeld, N. A., and Weigend, A. S., 1993, “The Future of Time Series,” Time Series Prediction: Forecasting the Future and Understanding the Past, Addison-Wesley, Reading, MA, 1–70.
12.
Holmes
P. J.
Moon
F. C.
1983
, “Strange Attractors and Chaos in Nonlinear Mechanics
,” ASME JOURNAL OF APPLIED MECHANICS
, Vol. 50
, pp. 1021
–1032
.13.
Kumar, A., and Mullick, S. K., 1995, “A Dynamical Systems Approach to Nonlinear Time Series Analysis for Modelling and Prediction,” preprint.
14.
Masri
S. F.
Miller
R. K.
Saud
A. F.
Caughey
T. K.
1987
, “Identification of Nonlinear Vibrating Structures: Part I—Formulation
,” ASME JOURNAL OF APPLIED MECHANICS
, Vol. 54
, pp. 918
–922
.15.
Moon, F. C., 1987, Chaotic Vibrations, John Wiley and Sons, New York.
16.
Moon
Y.
1995
, “Estimation of Mutual Information Using Kernel Density Estimators
,” Physical Review E
, Vol. 52
, pp. 2318
–2321
.17.
Ravindra
B.
Mallik
A. K.
1994
, “Role of Nonlinear Dissipation in Soft Duffing Oscillators
,” Physical Review E
, Vol. 49
, pp. 4950
–4954
.18.
Ravindra, B., and Cartmell, M., 1996, “Dynamical Systems Approach to Identification of Vibrating Systems,” International Conference on Identification in Engineering Systems, M. I. Friswell, ed., Swansea, UK, pp. 570–583.
19.
Ruelle, D., 1978, Thermodynamic Formalism, Addison-Wesley, Reading, MA.
20.
Sauer
T., et al.
1991
, “Embedology
,” Journal of Statistical Physics
, Vol. 65
, pp. 579
–616
.21.
Schouten
J. C.
1994
a, “Estimation of the Dimension of a Noisy Attractor
,” Physical Review E
, Vol. 50
, pp. 1851
–1861
.22.
Schouten
J. C.
1994
b, “Maximum-Likelihood Estimation of the Entropy of an Attractor
,” Physical Review E
, Vol. 49
, pp. 126
–129
.23.
Takens, F., 1981, “Detecting Strange Attractors in Turbulence,” Lecture Notes in Mathematics, Vol. 898, Springer-Verlag, New York, pp. 366–381.
24.
Tong, H., 1990, Nonlinear Time Series: A Dynamical Systems Approach, Oxford University Press, Oxford, UK.
25.
Wang
Z.
Fang
T.
1986
, “A Time Domain Method for Identifying Modal Parameters
,” ASME JOURNAL OF APPLIED MECHANICS
, Vol. 53
, pp. 28
–32
.
This content is only available via PDF.
Copyright © 1998
by The American Society of Mechanical Engineers
You do not currently have access to this content.
