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Makarenko N. G. Time series from geometry and topology of spatio-temporal chaos. Izvestiya VUZ. Applied Nonlinear Dynamics, 2004, vol. 12, iss. 6, pp. 3-16. DOI: 10.18500/0869-6632-2004-12-6-3-16
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Russian
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Article
UDC:
517.938; 523.98
Time series from geometry and topology of spatio-temporal chaos
Autors:
Makarenko Nikolaj Grigorevich, Federal state budgetary institution of science Main (Pulkovo) astronomical Observatory of Russian Academy of Sciences
Abstract:
The transformation of geometry and topology of 2D patterns into scalar time series with the help of the mathematical morphology and computational topology methods are considered.The approaches are illustrated by the example of the solar magnetic field investigation.
Key words:
Reference:
- Plato. The Republic. In: W3.p1. Moskow: Misl’, 1971, 621 p.
- Packard NH, Crutchfield JP, Farmer JD, Shaw RS. Geometry from а time series. Phys. Rev. Lett. 1980;45:712–716.
- Takens F. Detecting strange attractors in turbulence. Lecture Notes in Math. 1981;898:366–381.
- Afraimovich BC, Reiman AM. Dimensions and entropies in multidimensional systems. In: Nonlinear Waves. Dynamics and Evolution. Moskow: Nauka, 1989. P. 238–262.
- Sauer T, Yorke JА, Casdagli М. Embedology. J. Statist. Phys. 1991;65:579–616. DOI: 10.1007/BF01053745.
- Eckmann JР, Ruelle D. Ergodic theory оh chaos and strange attractors. Rev. Mod. Phys. 1985;57(3):617–656.
- Gilmore R, Lefranc M. The Topology of Chaos: Alice in Stretch and Squeezeland. New York: Wiley; 2002. 495 p.
- Rарр РЕ, Schah TI, Mees AI. Models оh knowing and the investigation оf dynamical systems. Physica D. 1999;132:133–149.
- Grassberger P, Procaccia I. On the characterization of strange attractors. Phys. Rev. Lett. 1983;50(5):346–349.
- Parker TS, Chua LO. Practical Numerical Algorithms for Chaotic Systems. New York: Springer; 1989. 348 р.
- Ott E, Sauer Т, Yorke JA. Coping with Chaos: Analysis оf Chaotic Data аnd the Exploitation of Chaotic Systems. Hoboken: John Wiley and Sons; 1994. 432 p.
- Schreiber Т. Interdisciplinary application оf nonlinear time series methods. Phys. Rep. 1999;308(2):1-64. DOI: 10.1016/S0370-1573%2898%2900035-0.
- Small M, Tse CK. Optimal embedding: A modelling paradigm // Physica D. Vol. 194. 2004. P. 283–296.
- Muldoon M, MacKay RS, Broomhead DC, Huke JP. Topology from time series. Physica D. 1993;65(1-2):1–16.
- Lay Ying-Cheng, Ye N. Recent developments in chaotic time series analysis. Int. J. Bifurcation and Chaos. 2003;13(6):1383–1422.
- Ruelle D. Chaotic evolution and strange attractors. The statistical analysis of time series for deterministic nonlinear systems. Cambridge University Press, 1989;33(2):112. DOI: 10.1137/1033084.
- Farmer JD, Sidorovich JJ. Predicting chaotic time series. Phys. Rev. Lett. 1987;59:845–848.
- Malinetskiy GG, Potapov AB. Modern problems of nonlinear dynamics. Moskow:URSS, 2002.P.358.
- Makarenko NG. Embedology and neuroprognosis. Lectures on neuroinformatics. In: V All-Russian Scientific and Technical Conference «Neuroinformatics-2003». Moskow:MIPHI; 2003. P. 86–148.
- Crutchfield JP, Kaneko K. Phenomenology оf Spatiotemporal Chaos. Directions in Chaos. Singapore:World Scientific; 1987. P. 272–353. DOI:10.1142/9789814415712_0008.
- Mayer-Kress G, Kaneko K. Spatiotemporal Chaos and Noise. J. Stat. Phys. 1989;54(5-6):1489–1508.
- Rabinovich MI, Fabrikant AP, Tsimring LS. Finite-dimensional spatial disorder. Nizhny Novgorod:UFN. 1992;35(8):629–649. DOI:10.1070/PU1992v035n08ABEH002253.
- Parlitz U, Merkwirth СА. Time series Analysis оf spatially extended systems. Intern. Symp. on Nonlinear Theory and its Applications «NOLTA’98». 14-17 September, 1998, Crans-Montana, Switzerland, P. 775–778.
- Michielsen K, Raedt HD. Morphological Image Analysis. Comp. Phys. Commun. 2000;132(1-2):94–103.
- Serra J. Image analysis and mathematical morphology. London:Academ. Press, 1988.610 p.
- Santalo L. Integral geometry and geometric probabilities. Moskow:Nauka, 1983.P.358.
- Thorpe J. Initial Chapters of Differential Geometry. Moscow: Mir, 1982.P.360.
- Adler R.J. The geometry оf random fields. Hoboken: John Wiley and Sons; 1981. 280 p.
- Stoyan D, Kendall WS, Mecke K. Stochastic Geometry and its applications. Hoboken: John Wiley and Sons; 1995. 436 p.
- Worsley KJ. The geometry of random images. Chance. 1996;9(1):27–40.
- Worsley K.J. Estimating the number оf peaks in а random field using е Hadwiger characteristic of excursion sets, with applications to medical images. Annals of Statistics. 1995;23(2): 640–669. DOI: 10.1214/aos/1176324540.
- Rosenfeld А, Klette R. Digital geometry. In: Proceedings of the 6th Joint Conference on Information Science. 8-13 March 2002, 11(3):30–33. DOI: 10.1007/BF03025195.
- Falconer K. Fractal geometry: Mathematical Foundations and Applications. Hoboken: John Wiley and Sons; 1990. 288 p.
- Mandelbrot B. The Fractal geometry of nature. Moskow:IKI, 2002. P. 654.
- Costa LdF, Kaye ВН, Montagnoli С. Accurate Fractal Estimation using Exact Dilations. In: Electronic Letters. 1999;(35): 1829–1836.
- Dey ТK, Edelsbrunner H, Guha S. Computational Topology. In: Advances in Discrete and Computational Geometry. Chazelle B, Goodman JE, R. Pollack R, editors. Contemporary Mathematics, AMS, Providence. 1998;223:109–144.
- Kaczynski Т, Mischaikow Т, Mrozek M. Computing Homology. Boston: Homology, Homotopy and Applications. 2001;5(2):233–256. DOI: 10.4310/HHA.2003.v5.n2.a8.
- Carlsson E, Carlsson G, de Silva V. An algebraic topological method for feature identification. International Journal of Computational Geometry & Applications. 2006;16(4):291–314. DOI:10.1142/S021819590600204X.
- Kelly JL. General Topology. Moskow:Nauka, 1968.432 p.
- Robins V, Meiss JD, Bradley Е. Computing connectedness: Disconnectedness and discreteness. Physica D. 2000;139:276–300.
- Robins V, Meiss JD, Bradley Е. Computing connectedness: An exercise in computational topology. Nonlinearity. 1998;11:913–922.
- Preparata F, Seamos M. Computational geometry. Introduction. Moskow: Mir, 1989. 478 p.
- Makarenko NG. How to obtain time series from geometry and topology of spatial patterns. Lectures on Neuroinformatics. Ch. 2. In: VI All-Russian Scientific and Technical Conference «Neuroinformatics-2004». Moskow:MIPHI; 2004. P. 140–199.
- Shapiro IS, Olshanetsky MA. Lectures on topology for physicists. Moscow-Izhevsk: RHD. 2001.128 р.
- Robins V. Computational Topology for Point Data: Betti Numbers of Alpha-Shapes. Morphology of Condensed Matter: Physics and Geometry of Spatially Complex Systems. In: Lecture Notes in Physics 600. Mecke K, Stoyan D, editors. Berlin: Springer. 2002. P. 261–274.
- Gamerio M, Kalies WD, Mischaikow K. Topological Characterization оf Spatial Temporal Chaos. Phys. Rev. E. 2004;70(3). DOI: https://doi.org/10.1103/PhysRevE.70.035203.
- Vitinsky YM, Kopetsky M, Kuklin GV. Statistics of Sunspotting Activity. Moscow: Nauka. 1986.P.296.
- Makarov VI, Tavastsherna KI. Global peculiarities of the process of solar activity. In: Variations of global characteristics of the Sun. Kiev: Naykova Dumka. 1992.P.270–301.
- Mouradian Z, Soru-Escaut I. Оn the dynamics оf the large-scale magnetic fields оf the Sun and the sunspot cycle. France: Astron. & Astroph. 1991;251:649–654.
- Aimanova GK, Makarenko N., Makarov VI, Tavastsherna KC. Estimation of order parameters of background solar magnetic fields from H-alpha maps. Period: 1914-1984. Solar Data. 1982;2:97–102.
- Makarenko N.G., Karimova L.M., Makarov V.I., Tavastsherna K.C. Contour statistics of large-scale solar fields. In: Modern Problems of Solar Cyclicity. St. Petersburg. 1997.P.139–143.
- Makarenko NG, Karimova LM, Novak M. Dynamics оf Solar magnetic fields from Synoptic charts. In: Emergent Nature. Patterns, Growth and Scaling in the Sciences. Singapore: World Scientific. 2001.P.197–207.
- Makarenko NG. Geometry and topology of random fields in solar physics. Studies on geomagnetism, aeronomy and solar physics. 2001;113:202–213.
- Dics С. Estimating invariants of noisy attractors. Phys.Rev.Е. 1996;53(5):P. R4263–R4266. DOI: https://doi.org/10.1103/PhysRevE.53.R426.
- Mundt MD. Maguire II WB, Chase RPR. Chaos in the sunspot Cycles: analysis and prediction. J.Geophys.Res. 1991;96(A2):1705–1716. DOI: 10.1029/90JA02150.
- Serre T, Nesme-Ribes Е. Nonlinear analysis оf solar cycles. Astron. Astrophys. 2000;360:319–330.
- Mordvinov AV, Salakhutdinova II, Plyusnina LA., Makarenko NG, Karimova LM. The topology of background magnetic fields and solar flare activity. Solar Physics. 2002;211:241–253. DOI: 10.1023/A:1022492003881.
- Lawrence JK, Cadavid AC, Ruzmaikin AA. On the multifractal distribution of Solar magnetic fields. Astrophys.J. 1996;465:425–435. DOI: 10.1086/177430
- Riedi RH. Multifractal Processes. In: Long range dependence: theory and applications. Doukhan, Oppenheim, Taqqu, editors. Switzerland: Birkhauser. 2002.P.625–715.
- Makarenko NG, Karimova L. Analysis of the global magnetic field of the Sun by methods of mathematical morphology and computational topology. Physics of the Sun and Stars. Elista. 2003.P.1–59.
Received:
20.12.2004
Accepted:
25.04.2005
Published:
15.06.2005
Journal issue:
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