Abstract
Various questions about the invariant measures of a dynamical system can be answered by computations of regular functionals or by ranking methods based on a set of observations. This includes symmetry tests and the determination of dimension coefficients. The paper contains the theoretical results and several simulations explain the methods.
Similar content being viewed by others
References
B. Bunow and G. H. Weiss,Math. Biosci. 47:221–237 (1979).
R. C. Bradley,Stock. Proc. Appl. 14:67–77 (1983).
I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai,Ergodic Theory. Grundl. der math. Wiss. 245, (Springer-Verlag, Berlin, 1982).
M. Denker,Mathem. Forsch. 12:35–47 (1982).
M. Denker and G. Keller,Z. Wahrscheinlichkeitstheorie verw. Geb. 64:505–522 (1983).
J. D. Farmer, E. Ott, and J. A. Yorke,Physica 7D:153–180 (1983).
P. Frederickson, J. L. Kaplan, E. D. Yorke, and J. A. Yorke,J. Diff. Eq. 49:185–207 (1983).
P. Grassberger and I. Procaccia,Phys. D 9:189–208 (1983).
H. S. Greenside, A. Wolf, J. Swift, and T. Pignataro,Phys. Rev. A(3) 25:3453–3456 (1982).
J. Guckenheimer,Comm. Math. Phys. 70:133–160 (1979).
I. Gumowski and Ch. Mira,Recurrences and Discrete Dynamic Systems, Lecture Notes in Mathematics No. 809 (Springer-Verlag, Berlin, 1980).
G. Györgyi and P. Szépfalusy,J. Slat. Phys. 34:451–475 (1984).
G. Györgyi and P. Szépfalusy,Z. Physik B 55:179–186 (1984).
E. J. Hannan,Time Series Analysis (Methuen & Co., London, 1960), chap. II.3.
F. Hofbauer and G. Keller,Math. Zeitschrift 180:119–140 (1982).
F. Hofbauer and G. Keller,Ergod. Th. Dyn. Syst. 2:23–43 (1982).
I. A. Ibragimov and Y. V. Linnik,Independent and Stationary Sequences of Random Variables (Wolters-Nordhoff Publishers, Groningen (NL), 1971).
J. L. Kaplan and J. A. Yorke,Functional Differential Equations and Approximation of Fixed Points, H. O. Peitgen and H. O. Walther, eds.,Springer Lect. Note. Math. 730:223–237 (1979).
G. Keller, Spectral analysis of fully developed chaotic maps, unpublished notes (1984).
G. Keller,Z. Wahrscheinlichkeitstheorie verw. Geb. 69:461–478 (1985).
T. Y. Li and J. A. Yorke,Amer. Math. Month. 82:985–992 (1975).
E. N. Lorenz,J. Atmos. Sci. 20:130–141 (1963).
B. Mandelbrot,The Fractal Geometry of Nature (Freeman, San Francisco, 1982).
R. M. May,Science 186:645–647 (1974).
R. M. May,Nature 261:459–467 (1976).
D. Mayer,Comm. Math. Phys. 95(1):1–15 (1984).
D. Mayer and G. Roepstorff,J. Stat. Phys. 31:309–326 (1983).
M. Misiurewicz,Publ. Math. IHES 53:17–51 (1981).
S. E. Newhouse,Entropy and Smooth Dynamics, Lecture Notes in Physics No. 179 (Springer-Verlag, Berlin, 1983), 165–179.
H. E. Nusse, Chaos, yet no chance to get lost. Thesis, University of Utrecht (NL) (1983).
J. C. Oxtoby,Measure and Category, Graduate Texts in Mathematics, Vol. 3 (Springer-Verlag, Berlin, 1971).
W. Parry,Topics in Ergodic Theory (Cambridge University Press, Boston, 1981).
K. Petersen,Ergodic Theory (Cambridge University Press, Boston, 1983).
W. Philipp and W. Stout,Mem. Am. Math. Soc. 161 (1975).
R. H. Randles and D. A. Wolfe,Introduction to the Theory of Nonparametric Statistics (Wiley, New York, 1979).
C. Sparrow,The Lorenz Equations: Bifurcations, Chaos and Strange Attractors in Applied Mathematical Sciences Vol. 41 (Springer-Verlag, Berlin, 1982).
W. Szlenk, I,Bol. Soc. Math. Mexico 24:57–82 (1979); II, Ergodic Theory and Dynamical Systems II (College Park, Maryland, 1979/1980), p. 75–95;Prog. Math. 21 (1982).
V. A. Volkonski and Y. A. Rozanov,Theory Prob. Appl. 6:186–198 (1961).
P. Walters,An Introduction to Ergodic Theory (Springer-Verlag, Berlin, 1982).
G. B. Wetherill,Intermediate Statistical Methods (Chapman and Hall, London/New York, 1981).
K. Yoshihara,Z. Wahrscheinlichkeitstheorie verw. Geb. 35:237–252 (1976).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Denker, M., Keller, G. Rigorous statistical procedures for data from dynamical systems. J Stat Phys 44, 67–93 (1986). https://doi.org/10.1007/BF01010905
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01010905