Abstract
Certain deterministic nonlinear systems may show chaotic behavior. We consider the motion of qualitative information and the practicalities of extracting a part from chaotic experimental data. Our approach based on a theorem of Takens draws on the ideas from the generalized theory of information known as singular system analysis. We illustrate this technique by numerical data from the chaotic region of the chaotic experimental data. The method of the singular-value decomposition is used to calculate the eigenvalues of embedding space matrix. The corresponding concrete algorithm to calculate eigenvectors and to obtain the basis of embedding vector space is proposed in this paper. The projection on the orthogonal basis generated by eigenvectors of timeseries data and concrete paradigm are also provided here. Meanwhile the state space reconstruction technology of different kinds of chaotic data obtained from dynamical system has also been discussed in detail.
Similar content being viewed by others
References
Packard, Crutchifield JP, Farmer JD, Shaw RS.Phys Rev Lett, 1980, 45: 712
Takens F et al. Dynamical systems of turbulence. In: Rand DA and Young IS, eds. Lecture Notes in Mathematics. Vol. 899. Berlin: Springer, 1986. 366
Ma Junhai, Chen Yushu, Liu Zengrong. For critical value influence studying of the different distributed phase-randomized to the data obtained in dynamic analysis.Applied Mathematics and Mechanics, 1998, 19(11): 1033–1042
Broomhead DS, King GP. Extracting qualitative dynamics from experimental data.Phys, 1987, D20: 217–236
Fraser AM. Reconstructing attractors from scalar time series: A comparison of singular system and redundancy criteria.Phys, 1989, D34: 391–404
Ma Junhai. The nonlinear dynamic system reconstruction of the chaotic timeseries. Ph D thesis, Tianjin University, China, May, 1997
Aldano AM, Muench J, Schwartz C: Singular-value decomposition and the G-P Algorithm.Phys Rev, 1988, A38: 3017–3026
Mees, AI, Rapp PE, Jennings LS. Singular-value decomposition and embedding dimension.Phys Rev, 1987, A36: 340–346
Franser AM, Swinney HL. Independent coordinates for strange attractors from mutual information.Phys Rev, 1986, A33: 1134–1140
Liebert W, Schuster HG. Proper choice of the time delay for the analysis of chaotic timeseries.Phys Lett, 1988, A142: 101–111
Casdagli M, Eubank S, Farmer JD, Gibson J. State space reconstruction in the presence of noise,Phys, 1991, D51: 52–98
Gibson JF, Farmer JD et al. An analytic approach to practical state space reconstruction.Phys, 1992, D57: 1–30.
Palus M, Albrecht V, Dvorak I. Information theoretic test for nonlinearity in timeseries.Phys Lett, 1993, A175: 203–209
Ma Junhai, Chen Yushu, Liu Zengrong. The discrimination criterion for identifying the random or nonlinear chaotic nature in timeseries.Applied Mathematics and Mechanics, 1998, 19(6): 481–488
Author information
Authors and Affiliations
Additional information
The project supported by the National Natural Science Foundation of China (19672043)
Rights and permissions
About this article
Cite this article
Yushu, C., Junhai, M. & Zengrong, L. The state space reconstruction technology of different kinds of chaotic data obtained from dynamical system. Acta Mech Sinica 15, 82–92 (1999). https://doi.org/10.1007/BF02487904
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02487904