Chaos theory in geophysics: past, present and future. (English) Zbl 1080.37610

The author suggests the usefulness of the ideas of the chaos theory in understanding geophysical phenomena. Through a systematic review of chaos theory applications in geophysics, with particular reference to the geophysical phenomena studied, promising results achieved and lessons learned, and potential areas for improvements, the author presents further support to such case. His analysis, based on the arguments that he himself put forth, is that “chaos theory” should not be viewed as a separate theory, but rather as a theory that “connects” both “deterministic” and “stochastic” theories, taking into account the advantages of the two theories as well as their limitations. The author believes that the ideas of chaos theory have immense potential in geophysics, even in the fields and problems not discussed in this article, and hopes that in the coming years and decades of this century would witness proof to such a belief.


37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
86A04 General questions in geophysics
Full Text: DOI


[1] Sivakumar, B; Berndtsson, R; Olsson, J; Jinno, K, Reply to “which chaos in the rainfall-runoff process”, Hydrol. sci. J, 47, 1, 149-158, (2002)
[2] Lorenz, E.N, Deterministic nonperiodic flow, J. atmos. sci, 20, 130-141, (1963) · Zbl 1417.37129
[3] Henon, M, A two-dimensional mapping with a strange attractor, Commun. math. phys, 50, 69-77, (1976) · Zbl 0576.58018
[4] May, R.M, Simple mathematical models with very complicated dynamics, Nature, 261, 459-467, (1976) · Zbl 1369.37088
[5] Rössler, O.E, An equation for continuous chaos, Phys. lett. A, 57, 397-398, (1976) · Zbl 1371.37062
[6] Mackey, M; Glass, L, Oscillations and chaos in physiological control systems, Science, 197, 287-289, (1977) · Zbl 1383.92036
[7] Lorenz, E.N, Atmospheric predictability as revealed by naturally occurring analogues, J. atmos. sci., 26, 636-646, (1969)
[8] Packard, N.H; Crutchfield, J.P; Farmer, J.D; Shaw, R.S, Geometry from a time series, Phys. rev. lett., 45, 9, 712-716, (1980)
[9] Takens, F, Detecting strange attractors in turbulence, (), pp. 366-81
[10] Grassberger, P; Procaccia, I, Measuring the strangeness of strange attractors, Physica D, 9, 189-208, (1983) · Zbl 0593.58024
[11] Grassberger, P; Procaccia, I, Characterisation of strange attractors, Phys. rev. lett, 50, 5, 346-349, (1983)
[12] Nicolis, C; Nicolis, G, Is there a climate attractor?, Nature, 311, 529-532, (1984)
[13] Grassberger, P, Do climate attractors exist?, Nature, 323, 609-612, (1986)
[14] Fraedrich, K, Estimating the dimensions of weather and climate attractors, J. atmos. sci., 43, 419-432, (1986)
[15] Fraedrich, K, Estimating weather and climate predictability on attractors, J. atmos. sci., 44, 722-728, (1987)
[16] Essex, C; Lookman, T; Nerenberg, M.A.H, The climate attractor over short time scales, Nature, 326, 64-66, (1987)
[17] Hense, A, On the possible existence of a strange attractor for the southern oscillation, Beitr. phys. atmos., 60, 1, 34-47, (1987)
[18] Tsonis, A.A; Elsner, J.B, The weather attractor over very short timescales, Nature, 333, 545-547, (1988)
[19] Wolf, A; Swift, J.B; Swinney, H.L; Vastano, A, Determining Lyapunov exponents from a time series, Physica D, 16, 285-317, (1985) · Zbl 0585.58037
[20] Grassberger, P; Procaccia, I, Estimation of the Kolmogorov entropy from a chaotic signal, Phys. rev. A, 28, 2591-2593, (1983)
[21] Farmer, D.J; Sidorowich, J.J, Predicting chaotic time series, Phys. rev. lett., 59, 845-848, (1987)
[22] Casdagli, M, Nonlinear prediction of chaotic time series, Physica D, 35, 335-356, (1989) · Zbl 0671.62099
[23] Sugihara, G; May, R.M, Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series, Nature, 344, 734-741, (1990)
[24] Casdagli, M, Chaos and deterministic versus stochastic nonlinear modeling, J. royal stat. soc. B, 54, 2, 303-328, (1991)
[25] Keppenne, C.L; Nicolis, C, Global properties and local structure of the weather attractor over western Europe, J. atmos. sci., 46, 2356-2370, (1989)
[26] Maasch, K.A, Calculating climate attractor dimension from δ_{18}O records by the grassberger – procaccia algorithm, Clim. dyn., 4, 45-55, (1989)
[27] Rodriguez-Iturbe, I; De Power, F.B; Sharifi, M.B; Georgakakos, K.P, Chaos in rainfall, Water resour. res., 25, 7, 1667-1675, (1989)
[28] Wilcox, B.P; Seyfried, M.S; Blackburn, W.H; Matison, T.H, Chaotic characteristics of snowmelt runoff: a preliminary study, ()
[29] Wilcox, B.P; Seyfried, M.S; Matison, T.H, Searching for chaotic dynamics in snowmelt runoff, Water resour. res., 27, 6, 1005-1010, (1991)
[30] Zeng, X; Pielke, R.A; Eykholt, R, Estimating the fractal dimension and the predictability of the atmosphere, J. atmos. sci., 48, 649-659, (1992)
[31] Islam, S; Bras, R.L; Rodriguez-Iturbe, I, A possible explanation for low correlation dimension estimates for the atmosphere, J. appl. meteor., 32, 203-208, (1993)
[32] Tsonis, A.A; Elsner, J.B; Georgakakos, K.P, Estimating the dimension of weather and climate attractors: important issues about the procedure and interpretation, J. atmos. sci., 50, 2549-2555, (1993)
[33] Zeng, X; Pielke, R.A, What does a low-dimensional weather attractor Mean?, Phys. lett. A, 175, 299-304, (1993)
[34] Jayawardena, A.W; Lai, F, Analysis and prediction of chaos in rainfall and stream flow time series, J. hydrol., 153, 23-52, (1994)
[35] Georgakakos, K.P; Sharifi, M.B; Sturdevant, P.L, Analysis of high-resolution rainfall data, (), 114-120
[36] Wang, Q, Correlation dimension estimates of global and local temperature data, J. appl. meteor., 34, 2556-2564, (1995)
[37] Jeong, G.D; Rao, A.R, Chaos characteristics of tree ring series, J. hydrol., 182, 239-257, (1996)
[38] Koutsoyiannis, D; Pachakis, D, Deterministic chaos versus stochasticity in analysis and modeling of point rainfall series, J. geophys. res, 101, D21, 26441-26451, (1996)
[39] Porporato, A; Ridolfi, L, Clues to the existence of deterministic chaos in river flow, Int. J. mod. phys. B, 10, 1821-1862, (1996)
[40] Puente, C.E; Obregon, N, A deterministic geometric representation of temporal rainfall: results for a storm in Boston, Water resour. res., 32, 9, 2825-2839, (1996)
[41] Sangoyomi, T.B; Lall, U; Abarbanel, H.D.I, Nonlinear dynamics of the great salt lake: dimension estimation, Water resour. res., 32, 1, 149-159, (1996)
[42] Waelbroeck, H; Lopex-Pena, R; Morales, T; Zertuche, F, Prediction of tropical rainfall by local phase space reconstruction, J. atmos. sci., 51, 22, 3360-3364, (1994)
[43] Abarbanel, H.D.I; Lall, U, Nonlinear dynamics of the great salt lake: system identification and prediction, Climate dyn, 12, 287-297, (1996)
[44] Abarbanel, H.D.I; Lall, U; Moon, Y.I; Mann, M; Sangoyomi, T, Nonlinear dynamics and the great salk lake: a predictable indicator of regional climate, Energy, 21, 7/8, 655-666, (1996)
[45] Theiler, J; Eubank, S; Longtin, A; Galdrikian, B; Farmer, J.D, Testing for nonlinearity in time series: the method of surrogate data, Physica D, 58, 77-94, (1992) · Zbl 1194.37144
[46] Kennel, M.B; Brown, R; Abarbanel, H.D.I, Determining embedding dimension for phase space reconstruction using a geometric method, Phys. rev. A, 45, 3403-3411, (1992)
[47] Schreiber, T; Grassberger, P, A simple noise reduction method for real data, Phys. lett. A, 160, 411-418, (1991)
[48] Schreiber, T, Extremely simple nonlinear noise reduction method, Phys. rev. E, 47, 4, 2401-2404, (1993)
[49] Amritkar, R.E; Pradeep Kumar, P, Interpolation of missing data using nonlinear and chaotic system analysis, J. geophys. res., 100, D2, 3149-3154, (1995)
[50] Cao, L; Mees, A; Judd, K, Dynamics from multivariate time series, Physica D, 121, 75-88, (1998) · Zbl 0933.62083
[51] Berndtsson, R; Jinno, K; Kawamura, A; Olsson, J; Xu, S, Dynamical systems theory applied to long-term temperature and precipitation time series, Trends hydrol., 1, 291-297, (1994)
[52] Porporato, A; Ridolfi, L, Nonlinear analysis of river flow time sequences, Water resour. res., 33, 6, 1353-1367, (1997)
[53] Porporato, A; Ridolfi, L, Multivariate nonlinear prediction of river flows, J. hydrol., 248, 1-4, 109-122, (2001)
[54] Sivakumar, B; Liong, S.Y; Liaw, C.Y; Phoon, K.K, Singapore rainfall behavior: chaotic?, J. hydrol. engg. ASCE, 4, 1, 38-48, (1999)
[55] Sivakumar, B; Phoon, K.K; Liong, S.Y; Liaw, C.Y, A systematic approach to noise reduction in chaotic hydrological time series, J. hydrol., 219, 3-4, 103-135, (1999)
[56] Jayawardena, A.W; Gurung, A.B, Noise reduction and prediction of hydrometeorological time series: dynamical systems approach vs. stochastic approach, J. hydrol., 228, 242-264, (2000)
[57] Elshorbagy, A; Panu, U.S; Simonovic, S.P, Analysis of cross-correlated chaotic streamflows, Hydrol. sci. J., 46, 5, 781-794, (2001)
[58] Elshorbagy, A; Simonovic, S.P; Panu, U.S, Estimation of missing streamflow data using principles of chaos theory, J. hydrol., 255, 123-133, (2002)
[59] Elshorbagy, A; Simonovic, S.P; Panu, U.S, Noise reduction in chaotic hydrologic time series: facts and doubts, J. hydrol., 256, 147-165, (2002)
[60] Islam, M.N; Sivakumar, B, Characterization and prediction of runoff dynamics: a nonlinear dynamical view, Adv. water resour., 25, 2, 179-190, (2002)
[61] Jayawardena, A.W; Li, W.K; Xu, P, Neighborhood selection for local modeling and prediction of hydrological time series, J. hydrol., 258, 40-57, (2002)
[62] Phoon, K.K; Islam, M.N; Liaw, C.Y; Liong, S.Y, A practical inverse approach for forecasting of nonlinear time series, J. hydrol. engg. ASCE, 7, 2, 116-128, (2002)
[63] Sivakumar, B; Berndtsson, R; Olsson, J; Jinno, K; Kawamura, A, Dynamics of monthly rainfall-runoff process at the Göta basin: a search for chaos, Hydrol. Earth syst. sci., 4, 3, 407-417, (2000)
[64] Sivakumar, B; Berndttson, R; Olsson, J; Jinno, K, Evidence of chaos in the rainfall-runoff process, Hydrol. sci. J., 46, 1, 131-145, (2001)
[65] Sivakumar, B, A phase-space reconstruction approach to prediction of suspended sediment concentration in rivers, J. hydrol., 258, 1-4, 149-162, (2002)
[66] Sivakumar, B; Jayawardena, A.W, An investigation of the presence of low-dimensional chaotic behavior in the sediment transport phenomenon, Hydrol. sci. J., 47, 3, 405-416, (2002)
[67] Sivakumar, B, Rainfall dynamics at different temporal scales: a chaotic perspective, Hydrol. Earth syst. sci., 5, 4, 645-651, (2001)
[68] Sivakumar, B, Is a chaotic multi-fractal approach for rainfall possible?, Hydrol. process., 15, 6, 943-955, (2001)
[69] Sivakumar, B; Sorooshian, S; Gupta, H.V; Gao, X, A chaotic approach to rainfall disaggregation, Water resour. res., 37, 1, 61-72, (2001)
[70] Lambrakis, N; Andreou, A.S; Polydoropoulos, P; Georgopoulos, E; Bountis, T, Nonlinear analysis and forecasting of a brackish karstic spring, Water resour. res., 36, 4, 875-884, (2000)
[71] Lisi, F; Villi, V, Chaotic forecasting of discharge time series: a case study, J. am. water resour. assoc., 37, 2, 271-279, (2001)
[72] Sivakumar, B; Jayawardena, A.W; Fernando, T.M.G.H, River flow forecasting: use of phase-space reconstruction and artificial neural networks approaches, J. hydrol., 265, 1-4, 225-245, (2002)
[73] Sivakumar, B; Persson, M; Berndtsson, R; Uvo, C.B, Is correlation dimension a reliable indicator of low-dimensional chaos in short hydrological time series?, Water resour. res., 38, 2 10.1029/2001WR000333, 3.1-3.8, (2002)
[74] Ghilardi, P; Rosso, R, Comment on “chaos in rainfall”, Water resour. res., 26, 8, 1837-1839, (1990)
[75] Schertzer, D; Tchiguirinskaia, I; Lovejoy, S; Hubert, P; Bendjoudi, H, Which chaos in the rainfall-runoff process? A discussion on “evidence of chaos in the rainfall-runoff process” by sivakumar et al, Hydrol. sci. J., 47, 1, 139-147, (2002)
[76] Sivakumar, B, Chaos theory in hydrology: important issues and interpretations, J. hydrol., 227, 1-4, 1-20, (2000)
[77] Clifford, N.J, Hydrology: the changing paradigm, Prog. phys. geog., 26, 2, 290-301, (2002)
[78] Sharifi, M.B; Georgakakos, K.P; Rodriguez-Iturbe, I, Evidence of deterministic chaos in the pulse of storm rainfall, J. atmos. sci., 47, 888-893, (1990)
[79] Sivakumar, B; Liong, S.Y; Liaw, C.Y, Evidence of chaotic behavior in Singapore rainfall, J. am. water resour. assoc., 34, 2, 301-310, (1998)
[80] Schouten, J.C; Takens, F; van den Bleek, C.M, Estimation of the dimension of a noisy attractor, Phys. rev. E, 50, 3, 1851-1861, (1994)
[81] Liu, Q; Islam, S; Rodriguez-Iturbe, I; Le, Y, Phase-space analysis of daily streamflow: characterization and prediction, Adv. water resour., 21, 463-475, (1998)
[82] Wang, Q; Gan, T.Y, Biases of correlation dimension estimates of streamflow data in the Canadian prairies, Water resour. res., 34, 9, 2329-2339, (1998)
[83] Krasovskaia, I; Gottschalk, L; Kundzewicz, Z.W, Dimensionality of Scandinavian river flow regimes, Hydrol. sci. J., 44, 5, 705-723, (1999)
[84] Stehlik, J, Deterministic chaos in runoff series, J. hydrol. hydromech., 47, 4, 271-287, (1999)
[85] Pasternack, G.B, Does the river run wild? assessing chaos in hydrological systems, Adv. water. resour., 23, 3, 253-260, (1999)
[86] Liaw, C.Y; Islam, M.N; Phoon, K.K; Liong, S.Y; Pasternack, G.B, Comment on “does the river run wild? assessing chaos in hydrological systems” in adv, Water. resour., 24, 5, 575-578, (2001)
[87] Pasternack, G.B, Reply to “comment on ‘does river run wild? assessing chaos in hydrological systems’s” by pasternack, Adv. water resour., 24, 5, 578-580, (2001)
[88] Sivakumar, B; Berndtsson, R; Persson, M, Monthly runoff prediction using phase-space reconstruction, Hydrol. sci. J., 46, 3, 377-388, (2001)
[89] Sivakumar, B, Discussion on “analysis of cross-correlated chaotic streamflows” by elshorbagy et al, Hydrol. sci. J., 47, 3, 523-527, (2002)
[90] Sivakumar B. Comment on “Estimation of missing streamflow data using principles of chaos theory” by Elshorbagy et al. J Hydrol 2002c, submitted for publication
[91] Zhou, Y; Ma, Z; Wang, L, Chaotic dynamics of the flood series in the huaihe river basin for the last 500 years, J. hydrol., 258, 100-110, (2002)
[92] Abarbanel, H.D.I, Analysis of observed chaotic data, (1996), Springer-Verlag New York, USA · Zbl 0875.70114
[93] Mikosch, T; Wang, Q, A Monte-Carlo method for estimating the correlation dimension, J. stat. phys., 78, 799-813, (1995) · Zbl 1102.82319
[94] Kawamura, A; McKerchar, A.I; Spigel, R.H; Jinno, K, Chaotic characteristics of the southern oscillation index time series, J. hydrol., 204, 168-181, (1998)
[95] Lorenz, E.N, Dimension of weather and climate attractors, Nature, 353, 241-244, (1991)
[96] Mohan, T.R.K; Rao, J.S; Ramaswamy, R, Dimension analysis of climatic data, J. clim., 2, 1047-1057, (1989)
[97] Broomhead, D.S; King, G.P, Extracting qualitative dynamics from experimental data, Physica D, 20, 217-236, (1986) · Zbl 0603.58040
[98] Tsonis, A.A; Elsner, J.B, Dimension analysis of climatic data–comments, J. clim., 3, 1502-1505, (1990)
[99] Mohan, T.R.K; Rao, J.S; Ramaswamy, R, Reply to “dimension analysis of climatic data–comments”, J. clim., 3, 1502-1505, (1990)
[100] Kurths, J; Herzel, H, An attractor in a solar time series, Physica D, 25, 165-172, (1987) · Zbl 0618.58039
[101] Kurths, J; Ruzmaikin, A.A, On forecasting the sunspot numbers, Solar phys., 126, 407-410, (1990)
[102] Mundt, M.D; Maguire, W.B; Chase, R.R.P, Chaos in the sunspot cycle: analysis and prediction, J. geophys. res., 96, 1705-1716, (1991)
[103] Jinno, K; Xu, S; Berndtsson, R; Kawamura, A; Matsumoto, M, Prediction of sunspots using reconstructed chaotic system equations, J. geophys. res., 100, A8, 14773-14781, (1995)
[104] Scargle, J.D, An introduction to chaotic and random time series analysis, Int. J. imag. syst. technol., 1, 243-259, (1989)
[105] Waelbroeck, H, Deterministic chaos in tropical atmospheric dynamics, J. atmos. sci., 52, 13, 2404-2415, (1995)
[106] Srivastava, H.N; Sinha, K.C, Predictability of geophysical phenomena using deterministic chaos, Proceedings of the national Academy of sciences, India, section-A, LXVII, Part VI2, 305-345, (1997)
[107] Gelhar, L.W, Stochastic subsurface hydrology, (1993), Prentice-Hall New Jersey
[108] Beven, K, Towards a coherent philosophy for modeling the environment, Proc. roy. soc. lond. A, 458, 2465-2484, (2002) · Zbl 1009.62105
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.