Abstract
Nonlinear dynamical systems produce complex temporal or spatial patterns by stretching and folding regions of phase-space in an iterative way. The topological and metric properties of this process can be extracted from the chaotic signal by using symbolic dynamics. The underlying “grammatical rules” are systematically detected and arranged on a logic tree. Predictions on the set of possible outcomes of the system are made and compared with the observation. The discrepancy between the two, evaluated through a generalization of the information gain, characterizes the complexity of the source. As a result of this unfolding procedure, the dynamics is described as a sequence of deterministic paths (blocks of symbols) which appear at random in time, with given transition probabilities. Fast hierarchical evaluations of invariant measures, dimensions, entropies and Lyapunov exponents are obtained from the logic tree, considering lower and lower levels (i.e., increasingly long symbol-sequences). Power spectra are accurately reproduced with a limited number of short orbits. The analysis applies to any system: dissipative or conservative, hyperbolic or not, invertible or not.
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References
E.N. Lorenz, J.Atmos.Sci. 20, 130 (1963).
D. Ruelle and F. Takens, Comm.Math.Phys. 20, 167 (1971).
B.B. Mandelbrot, “The Fractal Geometry of Nature”, Freeman, San Francisco (1982).
J.P. Eckmann and D. Ruelle, Rev.Mod.Phys. 57, 617 (1985).
P. Grassberger, in proc. conf. on “Chaos in Astrophysics”, Palm Coast, Florida, 1984, J. Perdang et al. editors, Reidl, Dortrecht (1985).
Chaos and Complexity”, R. Livi, S. Ruffo, S. Ciliberto and M. Buiatti Eds., World Scientific, Singapore, 1988.
Proceedings of the conference on Complex Systems, Gwatt, Switzerland, 1988.
R. Badii, “Quantitative Characterization of Complexity and Predictability”, submitted for publication.
P. Grassberger, in ref7 and Wuppertal preprint B 9, 1988;
D. Zambella and P. Grassberger, Wuppertal preprint B 11, 1988.
A.N. Kolmogorov, Probl.Inform.Transm. 1, 1 (1965);
G. Chaitin, J. Assoc.Comp.Math. 13, 547 (1966).
A. Lempel and J. Ziv, IEEE Trans.Inform.Theory 22, 75 (1976).
L Procaccia, S. Thomae and C. Tresser, Phys.Rev. A35, 1884 (1987).
M.J. Feigenbaum, M.H. Jensen and I. Procaccia, Phys.Rev.Lett. 57, 1503 (1986).
M.H. Jensen, L.P. Kadanoff and I. Procaccia, Phys.Rev. A36, 1409 (1987).
R. Badii, Riv. Nuovo Cim. 12, N° 3, 1 (1989).
M.J. Feigenbaum, J.Stat.Phys. 52, 527 (1988).
D.R. Hofstadter, “Gödel, Escher, Bach: an Eternal Golden Braid”, Vintage Books, New York (1980).
P. Grassberger and H. Kantz, Phys.Lett. 113A, 235 (1985).
R. Badii and G. Broggi, “Hierarchies of Relations between Partial Dimensions and Local Expansion Rates in Strange Attractors”, this issue.
V.M. Alekseev and M.V. Yakobson, Phys.Rep. 75, 290 (1981).
G. Györgyi and P. Szépfalusy, Phys.Rev. A31, 3477 (1985).
A. Renyi, “Probability Theory”, North-Holland, Amsterdam (1970).
J.D. Farmer, E. Ott and J.A. Yorke, Physica 7D, 153 (1983).
C. Grebogi, E. Ott and J.A. Yorke, Phys.Rev.Lett. 48, 1507 (1982).
R. Badii, unpublished.
M. Hénon, Comm.Math.Phys. 50, 69 (1976).
P. Cvitanovie, G. Gunaratne and I. Procaccia, Phys.Rev. A38, 1503 (1988).
D. Auerbach, P. Cvitanovic, J.P. Eckmann, G.H. Gunaratne and I. Procaccia, Phys.Rev.Lett. 58, 2387 (1987).
C. Grebogi, E. Ott and J.A. Yorke, Phys.Rev. A37, 1711 (1988).
P. Grassberger, R. Badii and A. Politi, J.Stat.Phys. 51. 135 (1988).
M.A. Sepúlveda and R. Badii “Symbolic Dynamical Resolution of Power Spectra”, this issue.
A. Politi, Phys.Lett. A136, 374 (1989).
E. Ott, W. Withers and J.A. Yorke, J.Stat.Phys. 36, 687 (1984).
P. Collet and J.P. Eckmann, “Iterated Maps on the Interval as Dynamical Systems”, Birkhauser, Cambridge, MA (1980).
M.J. Feigenbaum, J.Stat.Phys. 46, 919 and 925 (1987).
J.D. Farmer and J.J. Sidorowich, Phys.Rev.Lett, 59, 845 (1987);
J.D. Farmer and J.J. Sidorowich, in “Evolution, Learning and Cognition”, Ed. Y.C. Lee, World Scientific, Singapore (1989);
J.P. Crutchfield and B.S. McNamara, Complex Systems 1, 417 (1987);
J. Cremers and A. Hübler, Z.Naturforsch. 42a, 797 (1987);
M. Casdagli, Physica D, to appear (1989).
M. Sano and Y. Sawada, Phys.Rev.Lett. 55, 1082 (1985);
J.P. Eckmann, S. Oliffson Kamphorst, D. Ruelle and S. Ciliberto, Phys.Rev. 34A, 4971 (1986).
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Badii, R. (1989). Unfolding Complexity in Nonlinear Dynamical Systems. In: Abraham, N.B., Albano, A.M., Passamante, A., Rapp, P.E. (eds) Measures of Complexity and Chaos. NATO ASI Series, vol 208. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-0623-9_46
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DOI: https://doi.org/10.1007/978-1-4757-0623-9_46
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