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## References

1
  The TISEAN software package is publicly available at http://www.mpipks-dresden.mpg.de/~tisean. The distribution includes an online documentation system.

2
  H. Kantz and T. Schreiber, ``Nonlinear Time Series Analysis''. Cambridge University Press, Cambridge (1997).

3
  T. Schreiber, Interdisciplinary application of nonlinear time series methods, Phys. Reports 308, 1 (1999).

4
  D. Kaplan and L. Glass, ``Understanding Nonlinear Dynamics'', Springer, New York (1995).

5
  E. Ott, ``Chaos in Dynamical Systems'', Cambridge University Press, Cambridge (1993).

6
  P. Bergé, Y. Pomeau, and C. Vidal, ``Order Within Chaos: Towards a deterministic approach to turbulence'', Wiley, New York (1986).

7
  H.-G. Schuster, ``Deterministic Chaos: An introduction''. Physik Verlag, Weinheim (1988).

8
  A. Katok and B. Hasselblatt ``Introduction to the Modern Theory of Dynamical Systems'', Cambridge University Press, Cambridge (1996).

9
  E. Ott, T. Sauer, and J. A. Yorke, ``Coping with Chaos'', Wiley, New York (1994).

10
  H. D. I. Abarbanel, ``Analysis of Observed Chaotic Data'', Springer, New York (1996).

11
  P. Grassberger, T. Schreiber, and C. Schaffrath, Non-linear time sequence analysis, Int. J. Bifurcation and Chaos 1, 521 (1991).

12
  H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. Sh. Tsimring, The analysis of observed chaotic data in physical systems, Rev. Mod. Phys. 65, 1331 (1993).

13
  D. Kugiumtzis, B. Lillekjendlie, n. Christophersen, Chaotic time series I, Modeling, Identification and Control 15, 205 (1994).

14
  D. Kugiumtzis, B. Lillekjendlie, n. Christophersen, Chaotic time series II, Modeling, Identification and Control 15, 225 (1994).

15
  G. Mayer-Kress, ed., ``Dimensions and Entropies in Chaotic Systems'', Springer, Berlin (1986).

16
  M. Casdagli and S. Eubank, eds., ``Nonlinear Modeling and Forecasting'', Santa Fe Institute Studies in the Science of Complexity, Proc. Vol. XII, Addison-Wesley, Reading, MA (1992).

17
  A. S. Weigend and N. A. Gershenfeld, eds., ``Time Series Prediction: Forecasting the future and understanding the past'', Santa Fe Institute Studies in the Science of Complexity, Proc. Vol. XV, Addison-Wesley, Reading, MA (1993).

18
  J. Bélair, L. Glass, U. an der Heiden, and J. Milton, eds., ``Dynamical Disease'', AIP Press (1995).

19
  H. Kantz, J. Kurths, and G. Mayer-Kress, eds., ``Nonlinear analysis of physiological data'', Springer, Berlin (1998).

20
  T. Schreiber, Efficient neighbor searching in nonlinear time series analysis, Int. J. Bifurcation and Chaos 5, 349 (1995).

21
  F. Takens, ``Detecting Strange Attractors in Turbulence'', Lecture Notes in Math. Vol. 898, Springer, New York (1981).

22
  T. Sauer, J. Yorke, and M. Casdagli, Embedology, J. Stat. Phys. 65, 579 (1991).

23
  M. Richter and T. Schreiber, Phase space embedding of electrocardiograms, Phys. Rev. E 58, 6392 (1998)

24
  M. Casdagli, S. Eubank, J. D. Farmer, and J. Gibson, State space reconstruction in the presence of noise, Physica D 51, 52 (1991).

25
  A. M. Fraser and H. L. Swinney, Independent coordinates for strange attractors from mutual information, Phys. Rev. A 33, 1134 (1986).

26
  B. Pompe, Measuring statistical dependences in a time series, J. Stat. Phys. 73, 587 (1993).

27
  M. Paluš, Testing for nonlinearity using redundancies: Quantitative and qualitative aspects, Physica D 80, 186 (1995).

28
  M. B. Kennel, R. Brown, and H. D. I. Abarbanel, Determining embedding dimension for phase-space reconstruction using a geometrical construction, Phys. Rev. A 45, 3403 (1992).

29
  http://hpux.cs.utah.edu/hppd/hpux/Physics/embedding-26.May.93

30
  http://www.zweb.com/apnonlin/

31
  I. T. Jolliffe, ``Principal component analysis'', Springer, New York (1986).

32
  D. Broomhead and G. P. King, Extracting qualitative dynamics from experimental data, Physica D 20, 217 (1986).

33
  W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, ``Numerical Recipes'', 2nd edn., Cambridge University Press, Cambridge (1992).

34
  R. Vautard, P. Yiou, and M. Ghil, Singular-spectrum analysis: a toolkit for short, noisy chaotic signals, Physica D 58, 95 (1992).

35
  A. Varone, A. Politi, and M. Ciofini, CO laser with feedback, Phys. Rev. A 52, 3176 (1995).

36
  R. Hegger and H. Kantz, Embedding of sequences of time intervals, Europhys. Lett. 38, 267 (1997).

37
  J. P. Eckmann, S. Oliffson Kamphorst, and D. Ruelle, Recurrence plots of dynamical systems, Europhys. Lett. 4, 973 (1987).

38
  M. Casdagli, Recurrence plots revisited, Physica D 108, 206 (1997).

39
  N. B. Tufillaro, P. Wyckoff, R. Brown, T. Schreiber, and T. Molteno, Topological time series analysis of a string experiment and its synchronized model, Phys. Rev. E 51, 164 (1995).

40
  http://homepages.luc.edu/~cwebber

41
  A. Provenzale, L. A. Smith, R. Vio, and G. Murante, Distinguishing between low-dimensional dynamics and randomness in measured time series, Physica D 58, 31 (1992).

42
  H. Tong, ``Threshold Models in Non-Linear Time Series Analysis'', Lecture Notes in Statistics Vol. 21, Springer, New York (1983).

43
  A. Pikovsky, Discrete-time dynamic noise filtering, Sov. J. Commun. Technol. Electron. 31, 81 (1986).

44
  G. Sugihara and R. May, Nonlinear forecasting as a way of distinguishing chaos from measurement errors in time series, Nature 344, 734 (1990); Reprinted in [9].

45
  J.-P. Eckmann, S. Oliffson Kamphorst, D. Ruelle, and S. Ciliberto, Lyapunov exponents from a time series, Phys. Rev. A 34, 4971 (1986); Reprinted in [9].

46
  J. D. Farmer and J. Sidorowich, Predicting chaotic time series, Phys. Rev. Lett. 59, 845 (1987); Reprinted in [9].

47
  D. Auerbach, P. Cvitanović, J.-P. Eckmann, G. Gunaratne, and I. Procaccia, Exploring chaotic motion through periodic orbits, Phys. Rev. Lett. 58, 2387 (1987).

48
  O. Biham and W. Wenzel, Characterization of unstable periodic orbits in chaotic attractors and repellers, Phys. Rev. Lett. 63, 819 (1989).

49
  P. So, E. Ott, S. J. Schiff, D. T. Kaplan, T. Sauer, and C. Grebogi, Detecting unstable periodic orbits in chaotic experimental data, Phys. Rev. Lett. 76, 4705 (1996).

50
  P. Schmelcher, and F. K. Diakonos, A general approach to the finding of unstable periodic orbits in chaotic dynamical systems, Physical Review E 57, 2739 (1998).

51
  D. Kugiumtzis, O. C. Lingjærde, and N. Christophersen, Regularized local linear prediction of chaotic time series, Physica D 112 (1998) 344.

52
  L. Jaeger and H. Kantz, Unbiased reconstruction of the dynamics underlying a noisy chaotic time series, CHAOS 6 (1996) 440.

53
  M. Casdagli, Chaos and deterministic versus stochastic nonlinear modeling, J. Roy. Stat. Soc. 54, 303 (1991).

54
  D. Broomhead and D. Lowe, Multivariable functional interpolation and adaptive networks, Complex Syst. 2, 321 (1988).

55
  L. A. Smith, Identification and prediction of low dimensional dynamics, Physica D 58, 50 (1992).

56
  M. Casdagli, Nonlinear prediction of chaotic time series, Physica D 35, 335 (1989); Reprinted in [9].

57
  E. J. Kostelich and T. Schreiber, Noise reduction in chaotic time series data: A survey of common methods, Phys. Rev. E 48, 1752 (1993).

58
  M. E. Davies, Noise reduction schemes for chaotic time series, Physica D 79, 174 (1994).

59
  T. Schreiber, Extremely Simple Nonlinear Noise Reduction Method, Phys. Rev. E 47, 2401 (1993).

60
  D. R. Rigney, A. L. Goldberger, W. Ocasio, Y. Ichimaru, G. B. Moody, and R. Mark, Multi-channel physiological data: Description and analysis (Data set B), in [17].

61
  P. Grassberger, R. Hegger, H. Kantz, C. Schaffrath, and T. Schreiber, On noise reduction methods for chaotic data, CHAOS 3, 127 (1993); Reprinted in [9].

62
  H. Kantz, T. Schreiber, I. Hoffmann, T. Buzug, G. Pfister, L. G. Flepp, J. Simonet, R. Badii, and E. Brun, Nonlinear noise reduction: a case study on experimental data, Phys. Rev. E 48, 1529 (1993).

63
  M. Finardi, L. Flepp, J. Parisi, R. Holzner, R. Badii, and E. Brun, Topological and metric analysis of heteroclinic crises in laser chaos, Phys. Rev. Lett. 68, 2989 (1992).

64
  A. I. Mees and K. Judd, Dangers of geometric filtering, Physica D 68 427 (1993).

65
  T. Schreiber and M. Richter, Nonlinear projective filtering in a data stream, to appear in Int. J. Bifurcat. Chaos (1999).

66
  M. Richter, T. Schreiber, and D. T. Kaplan, Fetal ECG extraction with nonlinear phase space projections, IEEE Trans. Bio-Med. Eng. 45, 133 (1998).

67
  J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57, 617 (1985).

68
  R. Stoop and J. Parisi, Calculation of Lyapunov exponents avoiding spurious elements, Physica D 50, 89 (1991).

69
  H. Kantz, A robust method to estimate the maximal Lyapunov exponent of a time series, Phys. Lett. A 185, 77 (1994).

70
  M. T. Rosenstein, J. J. Collins, C. J. De Luca, A practical method for calculating largest Lyapunov exponents from small data sets, Physica D 65, 117 (1993).

71
  M. Sano and Y. Sawada, Measurement of the Lyapunov spectrum from a chaotic time series, Phys. Rev. Lett. 55, 1082 (1985).

72
  J. Kaplan and J. Yorke Chaotic behavior of multidimensional difference equations In Peitgen, H. O. & Walther, H. O., editors, ``Functional Differential Equations and Approximation of Fixed Points'' Springer, New York (1987).

73
  P. Grassberger and I. Procaccia, Physica D 9, 189 (1983).

74
  T. Sauer and J. Yorke, How many delay coordinates do you need?, Int. J. Bifurcation and Chaos 3, 737 (1993).

75
  J. Theiler, J. Opt. Soc. Amer. A 7, 1055 (1990).

76
  H. Kantz and T. Schreiber, Dimension estimates and physiological data, CHAOS 5, 143 (1995); Reprinted in [18].

77
  P. Grassberger, Finite sample corrections to entropy and dimension estimates, Phys. Lett. A 128, 369 (1988).

78
  F. Takens, in: B. L. J. Braaksma, H. W. Broer, and F. Takens, eds., ``Dynamical Systems and Bifurcations'', Lecture Notes in Math. Vol. 1125, Springer, Heidelberg (1985).

79
  J. Theiler, Lacunarity in a best estimator of fractal dimension, Phys. Lett. A 135, 195 (1988).

80
  J. M. Ghez and S. Vaienti, Integrated wavelets on fractal sets I: The correlation dimension, Nonlinearity 5, 777 (1992).

81
  R. Badii and A. Politi Statistical description of chaotic attractors, J. Stat. Phys. 40, 725 (1985).

82
  J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer, Testing for nonlinearity in time series: The method of surrogate data, Physica D 58, 77 (1992); Reprinted in [9].

83
  T. Schreiber and A. Schmitz, Improved surrogate data for nonlinearity tests, Phys. Rev. Lett. 77, 635 (1996).

84
  J. Theiler, P. S. Linsay, and D. M. Rubin, Detecting nonlinearity in data with long coherence times, in [17].

85
  T. Schreiber, Constrained randomization of time series data, Phys. Rev. Lett. 80 (1998) 2105.

86
  T. Subba Rao and M. M. Gabr, ``An Introduction to Bispectral Analysis and Bilinear Time Series Models'', Lecture Notes in Statistics Vol. 24, Springer, New York (1984).

87
  C. Diks, J. C. van Houwelingen, F. Takens, and J. DeGoede, Reversibility as a criterion for discriminating time series, Phys. Lett. A 201, 221 (1995).

88
  T. Schreiber and A. Schmitz, Discrimination power of measures for nonlinearity in a time series, Phys. Rev. E 55, 5443 (1997).

89
  J. Kadtke, Classification of highly noisy signals using global dynamical models, Phys. Lett. A 203, 196 (1995).

90
  R. Manuca and R. Savit, Stationarity and nonstationarity in time series analysis, Physica D 99, 134 (1996).

91
  M. C. Casdagli, L. D. Iasemidis, R. S. Savit, R. L. Gilmore, S. Roper, and J. C. Sackellares, Non-linearity in invasive EEG recordings from patients with temporal lobe epilepsy, Electroencephalogr. Clin. Neurophysiol. 102, 98 (1997).

92
  T. Schreiber, Detecting and analysing nonstationarity in a time series using nonlinear cross predictions, Phys. Rev. Lett. 78, 843 (1997).

93
  R. Cawley and G. H. Hsu, Local-geometric-projection method for noise reduction in chaotic maps and flows, Phys. Rev. A. 46, 3057 (1992).

94
  H. Kantz, Quantifying the closeness of fractal measures, Phys. Rev. E 49, 5091 (1994).

95
  T. Sauer, A noise reduction method for signals from nonlinear systems, Physica D 58, 193 (1992).

96
  E. Aurell, G. Boffetta, A. Crisanti, G. Paladin, and A. Vulpiani, Predictability in the large: an extension of the concept of Lyapunov exponent, J. Phys. A 30, 1 (1997).
97
  T. Schreiber and A. Schmitz, Classification of time series data with nonlinear similarity measures, Phys. Rev. Lett. 79, 1475 (1997).

Next: About this document Up: Practical implementation of nonlinear Previous: Acknowledgments