“Chaotic time series. Part I. Estimation of some invariant properties in state-space”
Authors: Dimitris Kugiumtzis, Bjørn Lillekjendlie and Nils Christophersen,Affiliation: University of Oslo and SINTEF
Reference: 1994, Vol 15, No 4, pp. 205-224.
Keywords: Time series, non-linear systems, chaos, noise, reconstruction, dimension of attractor, Lyapunov exponents
Abstract: Certain deterministic non-linear systems may show chaotic behaviour. Time series derived from such systems seem stochastic when analyzed with linear techniques. However, uncovering the deterministic structure is important because it allows for construction of more realistic and better models and thus improved predictive capabilities. This paper describes key features of chaotic systems including strange attractors and Lyapunov exponents. The emphasis is on state space reconstruction techniques that are used to estimate these properties, given scalar observations. Data generated from equations known to display chaotic behaviour are used for illustration. A compilation of applications to real data from widely different fields is given. If chaos is found to be present, one may proceed to build non-linear models, which is the topic of the second paper in this series.
PDF (2645 Kb)
DOI: 10.4173/mic.1994.4.1DOI forward links to this article:
| [1] Bjørn Lillekjendlie, Dimitris Kugiumtzis and Nils Christophersen (1994), doi:10.4173/mic.1994.4.2 |
| [2] Ali Gholipour, Babak N. Araabi and Caro Lucas (2006), doi:10.1007/s11063-006-9021-x |
| [3] T. J. Dodd and C. J. Harris (2002), doi:10.1080/00207720210147070 |
| [4] Rainer Hegger, Holger Kantz and Thomas Schreiber (1999), doi:10.1063/1.166424 |
| [5] K. K. Phoon, M. N. Islam, C. Y. Liaw and S. Y. Liong (2002), doi:10.1061/(ASCE)1084-0699(2002)7:2(116) |
| [6] H. Carli, L.G.S. Duarte and L.A.C.P. da Mota (2013), doi:10.1016/j.cpc.2013.12.001 |
| [7] Masoud Mirmomeni, Caro Lucas, Masoud Shafiee, Babak N. Araabi and Elaheh Kamaliha (2009), doi:10.1007/s00521-009-0235-5 |
| [8] D Kugiumtzis (1996), doi:10.1016/0167-2789(96)00054-1 |
| [9] Ali Gholipour, Caro Lucas, Babak N. Araabi, Masoud Mirmomeni and Masoud Shafiee (2007), doi:10.1007/s00521-006-0062-x |
| [10] Milan Palu (2007), doi:10.1080/00107510801959206 |
| [11] L. Barbosa, L. Duarte, C. Linhares and L. da Mota (2006), doi:10.1103/PhysRevE.74.026702 |
| [12] D. Kugiumtzis (1997), doi:10.1016/S0167-2789(96)00292-8 |
| [13] JAVAD RAZJOUYAN, SHAHRIAR GHARIBZADEH, ALI FALLAH, OMID KHAYAT, MITRA GHERGHEREHCHI, HOSSEIN AFARIDEH and MEHDI MOGHADDASI (2012), doi:10.1142/S0218127412500435 |
| [14] M. P. Hanias, I. L. Giannis and G. S. Tombras (2010), doi:10.1063/1.3293133 |
| [15] T. Aasen, D. Kugiumtzis and S.H.G. Nordahl (1997), doi:10.1006/cbmr.1997.1441 |
| [16] Hong-guang Ma and Chong-zhao Han (2006), doi:10.1007/s11460-005-0023-7 |
| [17] Theodoros E. Karakasidis and Avraam Charakopoulos (2009), doi:10.1016/j.chaos.2008.07.020 |
| [18] M. Mirmomeni, C. Lucas, B.N. Araabi, B. Moshiri and M.R. Bidar (2011), doi:10.1049/iet-spr.2009.0278 |
| [19] K.K. Ang and C. Quek (2006), doi:10.1109/TNN.2006.875996 |
| [20] Marko Gosak, Andra Sto er, Rene Markovi , Jurij Dolen ek, Marko Marhl, Marjan Slak Rupnik and Matja Perc (2015), doi:10.1063/1.4926673 |
| [21] Feng Liu, Chai Quek and Geok See Ng (2005), doi:10.1109/IJCNN.2005.1555956 |
| [22] Li Xihai, Liu Daizhi, Su Juan and Zhao Ke (2004), doi:10.1109/ICOSP.2004.1442283 |
| [23] Runjie Liu, Zhan Hou and Jinyuan Shen (2009), doi:10.1109/IWCFTA.2009.65 |
| [24] M. P. Hanias, D. A. Karras and M. Mobarak (2009), doi:10.1109/IWSSIP.2009.5367787 |
| [25] A. Chodorek and R.R. Chodorek (2002), doi:10.1109/ECUMN.2002.1002088 |
| [26] Francesco Lisi and Vigilio Villi (2001), doi:10.1111/j.1752-1688.2001.tb00967.x |
| [27] X. Rosalind Wang, Joseph T. Lizier, Oliver Obst, Mikhail Prokopenko and Peter Wang (2008), doi:10.1007/978-3-540-77690-1_6 |
| [28] M. P. Hanias, H. E. Nistazakis and G. S. Tombras (2012), doi:10.1007/978-3-642-29329-0_4 |
| [29] Gianluca Mattarocci (2006), doi:10.2139/ssrn.928796 |
| [30] K. Kiran, K. C. Ajithprasad, V. M. Ananda Kumar and K. P. Harikrishnan (2021), doi:10.1007/s11207-021-01761-0 |
| [31] Naomi Cedeno, Genesis Carillo, Maria J. Ayala, Sebastian Lalvay and Saba Infante (2021), doi:10.1007/978-3-030-90241-4_25 |
| [32] Shuo-Wen Chang, Chung-Ling Chang, Long-Tin Li and Shih-Wei Liao (2020), doi:10.1109/ACCESS.2019.2932413 |
| [33] Alex Eric Yuan and Wenying Shou (2022), doi:10.7554/eLife.72518 |
| [34] Mahmoud Alipour and Seyed Mohammad Reza Hashemi Golpayegani (2023), doi:10.1016/j.imu.2023.101277 |
| [35] Liufei Yang, Wei Lin and Siyang Leng (2023), doi:10.1063/5.0144310 |
[1] Abarbanel, H. D. I. (1993). Nonlinearity and chaos at work, Nature, 364:672 - 673 doi:10.1038/364672a0
[2] Abarbanel, H. D. I., R. Brown, M. B. Kennel. (1991). Lyapunov exponents in chaotic systems: their importance and their evaluation using observed data, International Journal of Modern Physics 13, 5:1347 - 1375, 1991 doi:10.1142/S021797929100064X
[3] Abraham, N. B., A. M. Albano, et al. (1986). Calculating the dimension of attractors front small data sets, Physics Letters A, 114:217 -221, 1986 doi:10.1016/0375-9601(86)90210-0
[4] Arrowsmith, D.K., C. M. Place. (1990). An Introduction to Dynamical Systems, Cambridge University Press.
[5] Babloyantz, A. A. Destexhe. (1986). Low-dimensional chaos in an instance of epilepsy, Proceedings of the National Academy of Sciences of the United States of America, 83:3513 - 3517, 1986 doi:10.1073/pnas.83.10.3513
[6] Badii, R. C. Broggi. (1988). Measurement of the dimension spectrum f(a): Fixed-mass approach, Physics Letters A, 131:339 - 343, 1988 doi:10.1016/0375-9601(88)90784-0
[7] Benettin, O. L. Galgani. (1980). Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them, Part I: Theory, Part II: Numerical application. Meccanica, 15:9 - 30, 1980 doi:10.1007/BF02128236
[8] Bingham, S. M. Kot. (1989). Multidimensional trees, range searching, and a correlation dimension algorithm of reduced complexity, Physics Letters A. 140:327 - 330, 1989 doi:10.1016/0375-9601(89)90629-4
[9] Brandstater, A. H. Swinney. (1987). Strange attractors in weakly turbulent Couette-Taylor flow, Physical Review A. 35:2207 - 2220, 1987 doi:10.1103/PhysRevA.35.2207
[10] Briggs, K. (1987). Simple experiments in chaotic dynamics, American Journal of Physics, 55:1083-1089. 1987 doi:10.1119/1.15297
[11] Briggs, K. (1990). An improved method for estimating Lyapunov exponents of chaotic time series, Physics Letters A. 151(1-2):27-32, 1990 doi:10.1016/0375-9601(90)90841-B
[12] Brock, W.A., D. A. Hsieh B. LeBaron. (1991). Nonlinear Dynamics, Chaos, and Instability: Statistical Theory and Economic Evidence, Massachusetts Institute of Technology, 1991.
[13] Broomhead, D.S., R. Jones, G. P. King. (1987). Topological dimension and local coordinates from time series data, Journal of Physics A: Mathematical and General, 20:563 - 569, 1987 doi:10.1088/0305-4470/20/9/003
[14] Broomhead, D.S. G. P. King. (1986). Extracting qualitative dynamics front experimental data, Physica D, 20:217 - 236, 1986 doi:10.1016/0167-2789(86)90031-X
[15] Bryant, P., R. Brown, H. D. I. Abarbanel. (1990). Lyapunov exponents from observed time series, Physical Review Letters, 6.13:1523 - 1526. 1990 doi:10.1103/PhysRevLett.65.1523
[16] Casdagli, M. (1992). Chaos and deterministic versus stochastic nonlinear modeling, Journal of the Royal Statistical Society Series B-Methodological, 54:303 - 328, 1992.
[17] Cawley, R. A. L. Licht. (1986). Maximum likelihood method for evaluating correlation dimension, Lecture Notes in Physics, 278:. - 103, 1986.
[18] Crutchfield, J.P., J. D. Farmer, N.H. Packard. R. S. Shaw. (1986). Chaos, Scientific American., pages 38 - 49, 1986.
[19] Destexhe, A., J. A. Sepulchre. A. Babloyantz. (1988). A comparative study of the experimental quantification of deterministic chaos, Physics Letters A, 132:101 - 106, 1988 doi:10.1016/0375-9601(88)90262-9
[20] Ditto, W.L. L. M. Peroca. (1993). Mastering chaos, Scientific American, pages 62 - 68. 1993.
[21] Dvorak, I. J. Klaschka. (1990). Modification of the Grassberger-Procaccia algorithm for estimating the correlation exponent of chaotic systems with high embedding dimension, Physics Letters A, 145:225 - 231, 1990 doi:10.1016/0375-9601(90)90355-R
[22] Eckmann, J.P. D. Ruelle. (1985). Ergodic theory of chaos and strange attractors, Reviews of Modern Physics. 5.3:617 - 656, 1985 doi:10.1103/RevModPhys.57.617
[23] Eckmann, J.P. D. Ruelle. (1992). Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems, Physica D, 56:185 - 187, 1992 doi:10.1016/0167-2789(92)90023-G
[24] Eckmann, S.O., J. P. Kamphorst, D. Ruelle, S. Ciliberto. (1986). Lyapunov exponents from time series, Physical Review A, 34:4971 - 4979, 1986.
[25] Ellner, S., A. R. Gallant. D. P. McCaffrey, D. W. Nychka. (1991). Convergence rates and data requirements for Jacobian based estimates of Lyapunov exponents from data, Physics Letters A, 153:357 - 363, 1991 doi:10.1016/0375-9601(91)90958-B
[26] Essex, C., T. Lookman. M.A.H. Nerenberg. (1987). The climate attractor over short time scales, Nature, 326:64 - 66, 1987 doi:10.1038/326064a0
[27] Farmer, J.D. (1982). Chaotic attractors of an infinite-dimensional dynamic system, Physica D, 4:366 - 393, 1982 doi:10.1016/0167-2789(82)90042-2
[28] Farmer, J.D., E. Ott, J.A. Yorke. (1983). The dimension of chaotic attractors, Physica D, 7:153 - 180, 1983 doi:10.1016/0167-2789(83)90125-2
[29] Farmer, J.D. J.J. Sidorowich. (1991). Optimal shadowing and noise reduction, Physica D, 47:373 - 392, 1991 doi:10.1016/0167-2789(91)90037-A
[30] Feder, J. (1988). Fractals, Plenum Press, New York.
[31] Franaszek, M. (1989). Optimized algorithm for the calculation of correlation integrals, Physical Review A, General Physics, 39:5540, 1989 doi:10.1103/PhysRevA.39.5440
[32] Frank, O.W., T. Lookinan, C. Essex, M. A. H. Nerenberg. (1990). Chaotic time series analyses of epileptic seizures, Physica D, 46:427 - 438, 1990.
[33] Fraser, A.M. (1989). Reconstructing attractors from scalar time series: a comparison of singular system and redundancy criteria, Physica D. 34:391 - 104, 1989 doi:10.1016/0167-2789(89)90263-7
[34] Fraser, A.M. H. Swinney. (1986). Independent coordinates for strange attractors from mutual information, Physical Review A, 33:1134 - 1140, 1986 doi:10.1103/PhysRevA.33.1134
[35] Frederickson, P., J. L. Kaplan, E. D. Yorke, J. A. Yorke. (1983). The Lyapunov dimension of strange attractors, Journal of Differential Equations, 49:185-207, 1983 doi:10.1016/0022-0396(83)90011-6
[36] Ghil, M., M. Kimoto, J. D. Neelin. (1991). Nonlinear dynamics and predictability in the atmospheric sciences, Reviews of Geophysics, 29:46 - 55, 1991 doi:10.1038/323609a0
[37] Grassberger, P. (1986). Do climatic attractors exist?, Nature, 323:609 - 612.
[38] Grassberger, P. (1990). An optimized box-assisted algorithm for fractal dimensions, Physics Letters A, 148:63 - 68, 1990 doi:10.1016/0375-9601(90)90577-B
[39] Grassberger, P. (1991). Non-linear time sequence analysis, International Journal of Bifurcation and Chaos, 1:52, 1991 doi:10.1142/S0218127491000403
[40] Grassberger, P. I. Procaccia. (1983). Characterization of strange attractors, Physical Review Letters, 50:346, 1983 doi:10.1103/PhysRevLett.50.346
[41] Grassberger, P. I. Procaccia. (1983). Measuring the strangeness of strange attractors, Physica D, 9:189 - 208, 1983 doi:10.1016/0167-2789(83)90298-1
[42] Greenside, H.S., A. Wolf, J. Swift, T. Pignataro. (1982). Impracticality of a box counting algorithm for calculating the dimensionality of strange attractors, Physical Review A, General Physics, 25:3453 - 3456, 1982 doi:10.1103/PhysRevA.25.3453
[43] Guckenheimer, I. P. Holies. (1983). Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York. 1983.
[44] Hale, J.K. H. Kocak. (1991). Dynamics and Bifurcations, Springer-Verlag, New York.
[45] Hammel, S.M. (1990). A noise reduction method for chaotic systems, Physics Letters A. 146(89):421 - 428, 1990 doi:10.1016/0375-9601(90)90493-8
[46] Harding, A.K., T. Shinbrot, J.M. Cordes. (1990). A chaotic attractor in timing noise from the Vela pulsar, The Astrophys. 353:588 - 596, 1990 doi:10.1086/168648
[47] Haucke, H. R. Ecke. (1987). Mode-locking and chaos in Rayleigh-Benard convection, Physica D, 25:307 - 329, 1987 doi:10.1016/0167-2789(87)90106-0
[48] Hausdorff, F. (1919). Dimension and äusseres mass, Math. Annalen, 79:157, 1919 doi:10.1007/BF01457179
[49] Hense, (1987). On the possible existence of a strange attractor for the Southern Oscillation, Beitr. Phys. Alm., 60:34 -47, 1987.
[50] Ilentschel, H.G.E. I. Procaccia. (1983). The infinite number of generalized dimensions of fractals and strange attractors, Physica D, 8:435 - 444, 1983 doi:10.1016/0167-2789(83)90235-X
[51] Holzfuss, J. G. Mayer-Kress. (1986). An approach to error estimation in the application of dimension algorithms, In G. Mayer-Kress, editor, Dimensions and Entropies in Chaotic Systems, page 257. Springer-Verlag, Berlin, 1986.
[52] Jensen, R.V. R. Urban. (1984). Chaotic price behavior in a nonlinear cobweb model, Econ. Len., 15:235. 1984 doi:10.1016/0165-1765(84)90106-X
[53] Kaplan, D.T. L. Glass. (1992). Direct test, for determinism in a time series. Physical Review Letters, 68:427 - 430, 1992 doi:10.1103/PhysRevLett.68.427
[54] Kennel, M.B., R. Brown. H.D I. Abarbanel. (1992). Determining embedding dimension for phase-space reconstruction using a geometrical construction, Physical Review A. 45:3403 - 3411, 1992 doi:10.1103/PhysRevA.45.3403
[55] Kostelich, E.J. J.A. Yorke. (1988). Noise reduction in dynamical systems, Physical Review Letters A, 38:1649 - 1652. 1988.
[56] Kostelich, E.J. J. A. Yorke. (1990). Noise reduction: Finding the simplest dynamical system consistent with the data, Physica D. 41:183 - 196, 1990 doi:10.1016/0167-2789(90)90121-5
[57] Lichtenberg, A.J. M. A. Liebermann. (1992). Regular and Chaotic Dynamics, Springer-Verlag, New York, second edition. 1992.
[58] Lichen, W., K. Pavvelzik, H. G. Schuster. (1991). Optimal embeddings of chaotic attractors from topological considerations, Europhysics Letters, 1.6:521 - 526, 1991 doi:10.1209/0295-5075/14/6/004
[59] Liebovitch, S. T. Toth. (1989). A fast algorithm to determine fractal dimensions by box counting, Physica Letters A. 141:386 - 390. 1989 doi:10.1016/0375-9601(89)90854-2
[60] Lorenz, E.N. (1963). Deterministic nonperiodic flow, J. Atmos. Sci, 20:130. 1963 doi:10.1175/1520-0469(1963)020%3C0130:DNF%3E2.0.CO;2
[61] Mackey, M. L. Glass. (1977). Oscillation and chaos in physological control systems, Science, 197:287, 1977 doi:10.1126/science.267326
[62] Mandelbrot, B.B. (1974). Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier, J. Fluid Mech, 62:331 - 358, 1974 doi:10.1017/S0022112074000711
[63] Mandelbrot, B.B. (1982). The Fractal Geometry of Nature, Freeman, San Francisco, 1982.
[64] Mane, R. (1981). On the dimensions of the compact invariant sets of certain non-linear maps, In D. A. Rand and L. S. Young, editors. Dynamical Systems and Turbulence. Warwick 1980, Lecture Notes in Mathematics 898, pages 230- 242. Springer, Berlin, 1981 doi:10.1007/BFb0091916
[65] May, R.M. (1976). Simple mathematical models with very complicated dynamics, Nature, 261:459, 1976 doi:10.1038/261459a0
[66] Mayer-Kress, O. (1986). Dimensions and Entropies in Chaotic Systems, Springer-Verlag, Berlin, 1986.
[67] McCaffrey, D.F., S. Ellner, A. R. Gallant, D. W. Nychka. (1992). Estimating the Lyapunov exponent of a chaotic system with nonparametric regression, Journal of the American Statistical Association, 8.419:682 - 695, 1992 doi:10.2307/2290206
[68] Mitschke, F., M. Moller, W. Lange. (1988). Measuring filter chaotic signals, Physical Review A. General Physics, 37:4518, 1988 doi:10.1103/PhysRevA.37.4518
[69] Nerenberg, M.A.H. C Essex. (1990). Correlation dimension and systematic geometric effects, Physical Review A, 42:7065 - 7074, 1990 doi:10.1103/PhysRevA.42.7065
[70] Nicolis, C. G. Nicolis. (1984). Is there a climatic attractor?, doi:10.1038/311529a0
[71] Osborne, A.R. A. Provenzale. (1989). Finite correlation dimension for stochastic systems with power-law spectra, Physica D, 35:357 - 381, 1989 doi:10.1016/0167-2789(89)90075-4
[72] Oseledec, V.I. (1968). A multiplicative ergodic theorem, Lyapunov characteristic numbers of dynamical systems. Mans, Moscow Math. Soc., 19:197 - 231, 1968.
[73] Packard, N.H., J.P. Crutchfield, J.D. Farmer. R.S. Shaw. (1980). Geometry from a time series, Physical Review Letters, 45:712, 1980 doi:10.1103/PhysRevLett.45.712
[74] Pesin, Y.B. Math. (1977). Surveys, 32:55. 1977 doi:10.1070/RM1977v032n04ABEH001639
[75] Provenzale, A., L. A. Smith. R. Vie. G. Murante. (1992). Distinguishing between low- dimensional dynamics and randomness in measure time series, Physica D. 58:31 - 49, 1992 doi:10.1016/0167-2789(92)90100-2
[76] Ramsey, H.-J., J.B. Yuan. (1990). The statistical properties of dimension calculations using small data sets, Nonlinearity, 3:155 - 176, 1990 doi:10.1088/0951-7715/3/1/009
[77] Russell, D.A., J. D. Hanson, E. Ott. (1980). Dimension of strange attractors, Physical Review Letters, 45:1175 - 1178. 1980 doi:10.1103/PhysRevLett.45.1175
[78] Sano, M. Y. Sawada. (1985). Measurement of the Lyapunov spectrum from chaotic time series, Physical Review Letters, 55:1082 - 1085, 1985 doi:10.1103/PhysRevLett.55.1082
[79] Schuster, H.G. (1989). Deterministic Chaos: An Introduction, Verlagsgesellschaft. Weinheim.
[80] Shitibrot, I., C. Grebogi. E. Ott, J. A. Yorke. (1993). Using small perturbations to control chaos, Nature, 363:411 -417, 1993 doi:10.1038/363411a0
[81] Smale, S. (1963). Diffeomorphisms with many periodic points, In S. S. Cairns, editor, Differential and Combinatorial Topology. pages 63 - 80. Princeton University Press, Princeton. 1963.
[82] Smale, S. (1967). Differentiable dynamical systems, American Mathematical Society Bulletin, 73:747 - 817, 1967 doi:10.1090/S0002-9904-1967-11798-1
[83] Smith, L.A. (1988). Intrinsic limits on dimension calculations, Physics Letters A, 133:283 - 288, 1988 doi:10.1016/0375-9601(88)90445-8
[84] Smith, L.A., J.-D. Fournier, E. A. Spiegel. (1986). Lacunarity and intermittency in fluid turbulence, Physics Letters A. 114:465 - 468, 1986 doi:10.1016/0375-9601(86)90695-X
[85] Smith, R.L. (1992). Estimating dimension in noisy chaotic time-series, Journal of the Royal Statistical Society Series B- Methodological, 54:329 - 351. 1992.
[86] Sugihara, G. R.M. May. (1990). Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series, Nature, 344:734 - 741, 1990 doi:10.1038/344734a0
[87] Takens, F. (1981). Detecting strange attractors in turbulence, In D. A. Rand and L. S. Young, editors. Dynamical Systems and Turbulence. Warwick 1980, Lecture Notes in Mathematics 898. pages 366 - 381. Springer, Berlin, 1981 doi:10.1007/BFb0091924
[88] Temam, R. (1988). Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer Verlag, New York. 1988.
[89] Theiler, J. (1987). Efficient algorithm for estimating the correlation dimension from a set of discrete points, Physical Review A, 3.9:4456 - 4462. 1987 doi:10.1103/PhysRevA.36.4456
[90] Theiler, J. (1990). Estimating fractal dimension, J. Opt. Soc. Am. A, .6:1055 -071, 1990 doi:10.1364/JOSAA.7.001055
[91] Theiler, J. (1990). Statistical precision of dimension estimators, Physical Review A. 4.6:3038 - 3051, 1990 doi:10.1103/PhysRevA.41.3038
[92] Theiler, J. (1991). Some comments on the correlation dimension of 1/fa noise, Physical Letters A, 155:480 - 492, 1991 doi:10.1016/0375-9601(91)90651-N
[93] Tsonis, A.A. (1992). Chaos: From Theory to Applications, Plenum Press, New York.
[94] Ulam, S.M. J. Von Neumann. (1947). On combinations of stochastic and deterministic processes, Bull. Amer. Math. Soc., 53:1120, 1947.
[95] Vautard, R. M. Ghil. (1989). Singular spectrum analysis in nonlinear dynamics with applications to paleoclimatic time series, Physica D, 35:395 - 424, 1989 doi:10.1016/0167-2789(89)90077-8
[96] Whitney, H. (1936). Differentiable manifolds, Ann. Math. 37:645 - 680, 1936 doi:10.2307/1968482
[97] Wiggins, S. (1990). Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer- Verlag, New York. 1990.
[98] Wolf, A., J. Swift, H. Swinney, J. Vastano. (1985). Determining Lyapunov exponents from a time series, Physica D. 16:285 - 317, 1985 doi:10.1016/0167-2789(85)90011-9
[99] Young, L.S. (1982). Dimension, entropy and Lyapunov exponents, Ergodic Theory and Dynamical Systems, 2:109.
BibTeX:
@article{MIC-1994-4-1,
title={{Chaotic time series. Part I. Estimation of some invariant properties in state-space}},
author={Kugiumtzis, Dimitris and Lillekjendlie, Bjørn and Christophersen, Nils},
journal={Modeling, Identification and Control},
volume={15},
number={4},
pages={205--224},
year={1994},
doi={10.4173/mic.1994.4.1},
publisher={Norwegian Society of Automatic Control}
};