Abstract
The theoretical ground of the method for defining a geometric characteristic as minimal attractor embedding dimension m o on the basis of matrix decomposition for different types of dynamical systems is proposed. On the subset of chaotic attractor in Euclidean space R m a function z(m) is constructed. It defines a measure of topological instability of the attractor when enlarging state space dimension (Rm → Rm+1). The value of z(m) changes monotonously when enlarging m, but if m ≥ m 0 , then z(m) does not depend on m. The computer confirmation of the theoretical results is presented. The investigation of digital electrocardiosignals using local-topological analysis of chaotic attractor trajectories is carried out.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Nicolis, G., Prigogine, I.: Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations. John Wiley & Sons, New York etc. (1977)
Haken, H.: Information and Self-Organization: A Macroscopic Approach to Complex Systems. Springer Verlag, Berlin (1988)
Berge, P., Pomeau Y., Vidal, C.: Order in Chaos: Towards a Deterministic Approach to Turbulence. John Wiley & Sons, New York (1986)
Grassberger P., Procaccia, I.: Characterization of strange attractors. Phys. Rev. Lett. 50 (1983) 346–349
Dailjudenko, V.F., Krot, A.M.: Calculation of the minimal embedding dimension of a chaotic attractor on the basis of local topological analysis of phase trajectories. Computation Mathematics and Mathematical Physics. 37 (1997) 311–319
Krot, A.M., Minervina, H.B.: Biomedical signal processing based on estimation of minimal dimension embedding of chaotic attractor. In: Proc. 10th Mediterranean Electrotechnical Conf. (Melecon 2000), Vol. 2, Cyprus, May 29–31 (2000) 733–736
Krot, A.M.: The decomposition of vector functions in vector-matrix series into state-space of nonlinear dynamic system. In: Proc. X European Signal Processing Conference (EUSIPCO 2000), Vol. 3, Tampere, Finland, 4–8 September (2000) 2453–2456
Krot, A.M.: Matrix decompositions of vector functions and shift operators on the trajectories of a nonlinear dynamical system. Nonlinear Phenomena in Complex Systems, 4, N2 (2001) 106–115
Dailjudenko, V.F., Krot, A.M., Minervina, E.B.: Active biomedical media exploration by means of spectral analysis approaches and chaotic signal attractor trajectories investigation. In: Proc. Intern. Workshop “Models and Analysis of Vocal Emissions for Biomedical Applications”, Firenze, Italy, September 1–3 (1999) 157–162
Krot, A.M., Minervina, H.B.: Minimal attractor embedding dimension on the basis of matrix decomposition for biomedical signal processing. In: Proc. 2nd Intern. Workshop “Models and Analysis of Vocal Emissions for Biomedical Applications”, Firenze, Italy, September 13–15 (2001) 80–82
Kalman, R.E., Falb, P.L., Arbib M.A.: Topics in Mathematical System Theory, Mc Graw-Hill Book Co., New York (1969)
Krot, A.M.: On a class of discrete quasistationary linear dynamic systems. 35 (1990) 711–713 (a translation of Dokl. Akad. Nauk USSR, 313 (1990) 1376–1380
Krot A.M., Minervina, H.B.: Minimal attractor embedding dimension for discrete dynamic system using state-space method: theoretical ground. In: Proc. 6th IEEE Intern. Conf. on Electronics, Circuits and Systems (ICECS’ 99), Vol.2, Pafos, Cyprus, September 5–8 (1999) 941–944
Horn, R.A., Johnson, C.R.: Matrix Analysis, Cambridge University Press, Cambridge (1986)
Krot, A.M., Minervina, E.B.: Fast Algorithms and Programs for Digital Spectral Processing of Signals and Images. Nauka i Tekhnika, Minsk (1995) (in Russian)
Takens, F. Detecting strange attractors in turbulence. In: Dynamical Systems and Turbulence. Lecture Notes in Math. Vol. 898, Springer, Berlin (1981) 366–381
Bezerianos, A., Bountis, T., Papaioannou, G., Polydoropoulos, P.: Nonlinear time series analysis of electrocardiograms. Chaos, 5 (1995) 95–101.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Krot, A.M., Minervina, H.B. (2003). Restoration of Dynamical Systems Attractors and Estimation of Their Geometric Characteristics into State-Space. In: Kumar, V., Gavrilova, M.L., Tan, C.J.K., L’Ecuyer, P. (eds) Computational Science and Its Applications — ICCSA 2003. ICCSA 2003. Lecture Notes in Computer Science, vol 2667. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44839-X_44
Download citation
DOI: https://doi.org/10.1007/3-540-44839-X_44
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40155-1
Online ISBN: 978-3-540-44839-6
eBook Packages: Springer Book Archive