Abstract
It is not always possible to resolve the dimension of an attractor from a finite data set. The number of data points required depends on the structure of the attractor, the distribution of points on the attractor, and the precision of the data. If the chaotic component of a system’s behaviour is sufficiently small relative to its large scale motion, and if orbits seldom visit the region of the attractor with small scale fractal structure, any method will fail to resolve the attractor’s dimension. It is simple to construct abstract mathematical examples that present this behaviour. However, while these limitations should be explicity recognised, it should also be noted that a growing body of empirical experience suggests that experimentally encountered physical and biological systems do not invariably display these behaviours. It is possible to estimate reliably the dimension of these attractors with comparatively small data sets.
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
N.B. Abraham, A.M. Albano, B. Das, G. de Guzman, S. Yong, R.S. Gioggia, G.P. Puccioni and J.R. Tredicce, Phys. Lett. 114A: 217–221 (1985).
K. Aihara and G. Matsumoto, in “Chaos”, A.V. Holden, ed., 257–269, University Press, Manchester & Princeton (1986).
A.M. Albano, J. Abounadi, T.H. Chyba, C.E. Searle and S. Yong, J. Opt. Soc. Amer. 2B: 47–55 (1985).
A.M. Albano et al., in “Dimension and Entropies in Chaotic Systems. Quantification of Complex Behaviour”, G. Mayer-Kress, ed., 231240, Springer-Verlag, Berlin (1986).
D.S. Broomhead and G.P. King, Physica 2OD: 217–236 (1986a).
D.S. Broomhead and G.P. King, in “Nonlinear Phenomena and Chaos”, S. Sarkan, ed., Adam Hilger, Bristol (1986b).
W.E. Caswell and J.A. Yorke, in “Dimensions and Entropies in Chaotic Systems”, G. Mayer-Kress, ed., 123–136, Springer-Verlag, Berlin (1986).
T.F. Chan, ACM Trans. Math. Software 8: 84–88 (1982).
T.A.C.M. Claasen and W.F.G. Mecklenbrauker, Phillips J. Res. 35: 217–250 (1980).
T.A.C.M. Claasen and W.F.G. Mecklenbrauker, Phillips J. Res. 35: 276–300 (1980).
T.A.C.M. Claasen and W.F.G. Mecklenbrauker, Phillips J. Res. 35: 372–389 (1980).
J.-P. Eckmann and D. Ruelle, Rev. modn. Phys. 54: 617–656 (1985).
J.D. Farmer, in “Evolution of Order and Chaos”, H. Haken, ed., 228–246, Springer-Verlag, Berlin (1982a).
J.D. Farmer, Z. Naturforsch. 37A: 1304–1325 (1982b).
J.D. Farmer, E. Ott and J.A. Yorke, Physica 7D: 153–180 (1983).
A.M. Fraser and H.L. Swinney, Phys. Rev. 33A: 1134–1140 (1986).
L. Glass, A. Shrier and J. Belair, in “Chaos”, A.V. Holden, ed., 237–256, University Press, Manchester & Princeton (1986).
G. Golub and W. Kahan, SIAM J. Numer. Anal. 2: 205–224 ( (1965).
G.B. Golub and C. Reinsch, Numer. Math. 14: 403–420 (1970).
G.H. Golub and C.F. Van Loan, “Matrix Computations”, Johns Hopkins University Press, Baltimore (1983).
P. Grassberger and I. Procaccia, Physica 9D: 189–208 (1983a).
P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50: 346–349 (1983b).
P. Grassberger and I. Procaccia, Phys. Rev. A 28A: 2591–2593 (1983c).
M.R. Guevara, L. Glass and A. Shrier, Science, Wash. 214: 1350–1353 (1981).
H. Hayashi, S. Ishizuka and K. Hirakawa, Phys. Lett. A 98A: 474–476 (1983).
H. Hayashi, S. Ishizuka, M. Ohta and K. Hirakawa, Phys. Lett. 88A: 435–438 (1983).
M. Hénon, Commun. math. Phys. 50: 69–78 (1976).
H.G.E. Hentschel and I. Procaccia, Physica 8D: 435–444 (1983).
M.H. Jensen, L.P. Kadanoff, A. Libchaber, I. Procaccia and J. Stavans, Phys. Rev. Lett. 55: 2798–2801 (1986).
J. Kurth and H. Herzel, Physica D, submitted (1986).
M. Markus and B. Hess, Proc. natn. Acad. Sci. U.S.A. 81: 4394–4398 (1984).
M. Markus, D. Kuschmitz and B. Hess, FEBS Lett. 172: 235–238 (1984).
J.L. Martiel and A. Goldbeter, Nature, Lond. 313: 590–592 (1985).
A.I. Mees, P.E. Rapp and L.S. Jennings, “Singular value decomposition and embedding dimension”, Phys. Rev., in press.
N.H. Morgan and A.S. Gevins, IEEE Trans. Biomed. Eng. BME-33: 6670 (1986).
C. Nicolis and G. Nicolis, Nature, Lond. 311: 529–532 (1984).
P.E. Rapp, I.D. Zimmerman, A.M. Albano, G.C. de Guzman and N.N. Greenbaum, Phys. Lett. 110A: 335–338 (1985a).
P.E. Rapp, I.D. Zimmerman, A.M. Albano, G.C. de Guzman, N.N. Greenbaum and T.R. Bashore, in “Nonlinear Oscillations in Chemistry and Biology”, H.G. Othmer, ed., Springer-Verlag, NY (1985b).
W.M. Schaffer, IMA J. Maths. appl. Med. and Biol. 2: 221–252 (1985).
C.W. Simm, M.L. Sawley, F. Skiff and A. Pochelon, “On the analysis of experimental signals for evidence of determinstic chaos”, preprint (1986).
F. Takens, in “Dynamical Systems aid Turbulence, Lecture Notes in Mathematics, Volume 898”, D.A. Rand and L.S. Young, eds., 365381, Springer-Verlag, NY (1980).
J. Theiler, Phys. Rev. A 34: 2427 (1986).
J. Vastano and E.J. Kostelich, in “Dimensions and Entropies in Chaotic Systems”, G. Mayer-Kress, ed., 100–107, Springer-Verlag, Berlin (1986).
S. Watanabe, Proc. Conference on Information Theory, Prague (1965).
H. Whitney, Ann. Math. 37: 645–680 (1936).
E. Wigner, Phys. Rev. 40: 749–759 (1932).
A. Wolf, J.B. Swift, H.L. Swinney and J.A. Vastano, Physica 16D: 285–317 (1985).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer Science+Business Media New York
About this chapter
Cite this chapter
Albano, A.M., Mees, A.I., de Guzman, G.C., Rapp, P.E. (1987). Data Requirements for Reliable Estimation of Correlation Dimensions. In: Degn, H., Holden, A.V., Olsen, L.F. (eds) Chaos in Biological Systems. NATO ASI Series, vol 138. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-9631-5_24
Download citation
DOI: https://doi.org/10.1007/978-1-4757-9631-5_24
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-9633-9
Online ISBN: 978-1-4757-9631-5
eBook Packages: Springer Book Archive