Abstract
Topologically equivalent attractor reconstruction is one of the major issues in nonlinear analysis. This is because of the fact that the underlying dynamical model of some nonlinear phenomena may not be known and thus it is necessary to retrieve the dynamics from the data it generates. One way to achieve this is the reconstruction of the attractor. The basis of such reconstruction is the famous Taken’s embedding theorem, which asserts that an equivalent phase space trajectory,preserving the topological structures of the original phase space trajectory, can be reconstructed by using only one observation of the time series. However, in some cases topologically equivalent attractor reconstructions can also be done by using multiple observations. All these things involve the choice of suitable time-delay(s) and embedding dimension. Various measures are available to find out the suitable time-delay(s). Among them, linear auto-correlation, Average mutual information, higher dimensional mutual information are mostly used measures for the reconstruction of the attractors. Every measures have certain limitations in the sense that they are not always useful in finding suitable time-delay(s). Thus it is necessary to introduce few more nonlinear measures, which may be useful if the aforesaid measures fail to produce suitable time-delay/time-delays. In this chapter, some comparatively new nonlinear measures namely generalized auto-correlation, Cross auto-correlation and a new type of nonlinear auto-correlation of bivariate data for finding suitable time-delay(s) have been discussed. To establish their usefulness, attractors of some known dynamical systems have been reconstructed from their solution components with suitable time-delay(s) obtained by each of the measures. These attractors are then compared with their corresponding original attractor by a shape distortion parameter Sd. This shape distortion parameter actually checks how much distorted the reconstructed attractor is from its corresponding original attractor. The main objective of this chapter is to address the problem of reconstruction of a least distorted topologically equivalent attractor. The reason is that if the reconstructed attractor is least distorted from its original one, the dynamics of the system can be retrieved more accurately from it. This would help in identifying the dynamics of the corresponding system, even when the dynamical model is not known. Out of the three measures discussed in this chapter, the generalized and cross auto-correlation measures produce least distorted topologically equivalent attractor only by consideration of multiple solution components of the dynamical system. On the other hand, by using the measure—new type of nonlinear auto-correlation of bivariate data, one can reconstruct a least distorted topologically attractor from single solution component of the dynamical system. Various numerical results on Lorenz system, Neuro-dynamical system and also on two real life signals are presented to prove the effectiveness of the aforesaid three comparatively new nonlinear time-delay finding measures. Finding of suitable embedding dimension is another important issue for attractor reconstruction. However, this issue has not been highlighted in this chapter because we have restricted this discussion only to three dimensional attractor reconstruction.
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References
Williams, G.P.: Chaos Theory Tamed. Joseph Henry Press, Washington, DC (1997)
Kaplan, D.T., Glass, L.: Understanding Nonlinear Dynamics. Springer, New York (1995)
Abarbanel, H.D.I.: Analysis of Observed Chaotic Data. Springer, New York (1997)
Kaplan, D.T.: Signal Processing. Warwick-exercises.tex, 5–9 (2004)
Glass, L., Mackey, M.C.: From Clocks to Chaos: The Rhythms of Life. Princeton University Press, Princeton (1988)
Strogatz, S.H.: Non Linear Dynamics and Chaos-with Application to Physics, Biology, Chemistry and Engineering. Advance Book Program (Persueus Books), Cambridge (1994)
Banbrook, M., McLaughlin, S.: Is Speech Chaotic? Invariant Geometrical Measures for Speech Data. IEE Colloq. Exploit. Chaos Signal Process. 193, 1–8 (1994)
McLaughlin, S., Banbrook, M., Mann, I.: Speech characterization and synthesis by nonlinear methods. IEEE Trans. Speech Audio Process. 7, 1–17 (1999)
Johnson, T.L., Dooley, K.: Looking for chaos in time series data. In: Sulis, W., Combs, A. (eds.) Nonlinear Dynamics in Human Behavior, pp. 44–76. World Scientific, Singapore (1996)
Kugiumtzis, D.: State space reconstruction parameters in the analysis of chaotic time series’ the role of the time window length. Physica D 95, 13 (1996)
Takens, F.: Detecting strange attractors in turbulence. In: Rand, D.A., Young, L.S. (eds.) Dynamical Systems and Turbulence. Lecture Notes in Mathematics, vol. 898, pp. 366–381. Springer, Berlin (1981)
Sauer, T., Yorke, J., Casdagli, M.: Embedology. J. Stat. Phys. 65, 579–616 (1991)
Gibson, J.F., Farmer, J.D., Casdagli, M., Eubank, S.: An analytic approach to practical state space reconstruction. Physica D 57, 1–30 (1992)
Kantz, H., Olbrich, E.: Scalar observations from a class of high-dimensional chaotic systems: limitations of the time delay embedding. Chaos 7(3), 423 (1997)
Olbrich, E., Kantz, H.: Inferring chaotic dynamics from time-series: on which length scale determinism becomes visible. Phys. Lett. A 232, 63 (1997)
Pecora, L.M., Moniz, L., Nichols, J., et al.: A unified approach to attractor reconstruction. Chaos 17, 013110 (2007)
Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (1993)
Rosenstein, M.T., Collins, J.J., De Luca, C.J.: A practical method for calculating largest Lyapunov exponents from small data sets. Physica D 65, 117–134 (1993)
Wolf, A.: Quantity chaos with Lyapunov exponents. In: Holden, A.V. (ed.) Chaos, pp. 273–290. Princeton University Press, New Jersey (1986)
Benettin, B., Galgani, L., Giorgilli, A., Strelcyn, J.: Lyapunov characteristic exponents for smooth dynamical systems and for hamiltonian systems: a method for computing all of them Part I, II. Meccanica 15(1), 9–30 (1980)
Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Detennining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985)
Eckmann, J.P., Kamphorst, S.O., Ruelle, D., Ciliberto, S.: Lyapunov exponents from time series. Phys. Rev. A 34, 4971–4979 (1986)
Sano, M., Sawada, Y.: Measurement of the Lyapunov spectrum from a chaotictime series. Phys. Rev. Lett. 55, 1082–1085 (1985)
Peitgen, H., Jurgens, H., Saupe, D.: Chaos and Fractals-New Frontiers of Science. Springer, New York (1996)
Banbrook, M., Ushaw, G., McLaughlin, S.: How to extract Lyapunov exponents from short and noisy time series. IEEE Trans. Signal Process. 45, 1378–1382 (1997)
Geist, K., Parlitz, U., Lauterbom, W.: Comparison of different methods for computing Lyapunov exponents. Prog. Theor. Phys. 83, 875 (1990)
Kantz, H.: A robust method to estimate the maximal Lyapunov exponent of a time series. Phys. Lett. A 185, 77–87 (1994)
Kennedy, M.P.: Basic concepts of nonlinear dynamics and chaos. In: Toumazou, C. (ed.) Circuits and Systems Tutorials, pp. 289–313. IEEE Press, London (1994)
Briggs, K.: An improved method for estimating Liapunov exponents of chaotic time series. Phys. Lett. A 151, 27–32 (1990)
Packard, N.H., Crutchfield, J.P., Farmer, J.D., Shaw, R.S.: Geometry from a time series. Phys. Rev. Lett. 45, 712–716 (1980)
Abarbanel, H.D.I., Carroll, T.A., Pecora, L.M., Sidorowich, J.J., Tsimring, L.S.: Predicting physical variables in time delay embedding. Phys. Rev. E 49, 1840–1853 (1994)
Guckenheimer, J., Buzyna, G.: Dimension measurements for geostrophic turbulence. Phys. Rev. Lett. 51, 1438–1441 (1983)
Palus, M., Dvorák, I., David, I.: Spatio-temporal dynamics of human EEG. Physica A 185, 433–438 (1992)
Cao, L., Mees, A., Judd, K.: Dynamics from multivariate time series. Physica D 121, 75–88 (1998)
Barnard, J.P., Aldrich, C., Gerber, M.: Embedding of multidimensional time-dependent observations. Phys. Rev. E 64, 046201 (2001)
Garcia, S.P., Almeida, J.S.: Multivariate phase space reconstruction by nearest neighbor embedding with different time delays. Phys. Rev. E 72, 027205 (2005)
Hirata, Y., Suzuki, H., Aihara, K.: Reconstructing state spaces from multivariate data using variable time delays. Phys. Rev. E 74, 026202 (2006)
Grassberger, P., Procaccia, I.: Measuring the strangeness of strange attractors. Physica D 9, 189–208 (1983)
Kennel, M.B., Brown, R., Abarbanel, H.D.I.: Determining minimum embedding dimension using a geometrical construction. Phys. Rev. A 45, 3403–3411 (1992)
Cao, L.: Practical method for determining the minimum embedding dimension of a scalar time series. Physica D 110, 43–50 (1997)
Kember, G., Fowler, A.C.: A correlation function for choosing time delays in phase portrait reconstructions. Phys. Lett. A 179, 72–80 (1993)
Theiler, J.: Spurious dimension from correlation algorithms applied to limited time-series data. Phys. Rev. A 34, 2427–2432 (1986)
Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis. Cambridge University Press, Cambridge (1997)
Fraser, A.M., Swinney, H.L.: Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33, 1134–1140 (1986)
Liebert, W., Schuster, H.G.: Proper choice of time delay for the analysis of chaotic time series. Phys. Lett. A 142, 107 (1989)
Kraskov, A., Stögbauer, H., Grassberger, P.: Estimating mutual information. Phys. Rev. E 69(6), 066138 (2004)
Simon, G., Verleysen, M.: High-dimensional delay selection for regression models with mutual information and distance-to-diagonal criteria. Neurocomputing 70, 1265–1275 (2007)
Lorentz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963)
Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)
Vlachos, I., Kugiumtzis, D.: Non uniform state-space reconstruction and coupling detection. Phys. Rev. E 82, 016207 (2010)
Palit, S.K., Mukherjee, S., Bhattacharya, D.K.: New types of nonlinear auto-correlations of bivariate data and their applications. Appl. Math. Comput. 218(17), 8951–8967 (2012)
Palit, S.K., Mukherjee, S., Bhattacharya, D.K.: Generalized auto-correlation and its application in attractor reconstruction. Bull. Pure Appl. Math. 5(2), 218–230 (2011)
Palit, S.K., Mukherjee, S., Bhattacharya, D.K.: A high dimensional delay selection for the reconstruction of proper phase space with cross auto-correlation. Neurocomputing 113, 49–57 (2013)
Rulkov, N.F., Sushchik, M.M., Tsimring, L.S., Abarbanel, H.D.I.: Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E 51, 980–994 (1995)
Marchal, C.: Determinism, random, chaos, freedom. Henri Poincare and the revolution of scientific ideas in the twentieth century. Regul. Chaotic Dyn. 10, 227–236 (2005)
Wichard, J.D., Parltz, U., Lauterborn, W.: Application of nearest neighbor’s statistics. In: International Symposium on Nonlinear Theory and Its Applications, Switzerland (1998)
Martinez, W.L., Martinez, A.R.: Computational Statistics Handbook with Matlab. Chapman & Hall/CRC, New York (2002)
Theiler, J., Eubank, S., Longtin, A., Galdrikian, B., Farmer, J.: Testing for nonlinearity in time series: the method of surrogate data. Physica D 58, 77–94 (1992)
Schreiber, T., Schmitz, A.: Improved Surrogate Data for Nonlinearity Tests. Phys. Rev. Lett. 77, 635 (1996)
Jar, J.H.: Biostatistical Analysis. Pearson International Edition, Harlow (2006)
Das, A., Roy, A.B.: Chaos in a three dimensional neural network. Appl. Math. Model. 24, 511–522 (2000)
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Palit, S.K., Mukherjee, S., Banerjee, S., Ariffin, M.R.K., Bhattacharya, D.K. (2015). Some Time-Delay Finding Measures and Attractor Reconstruction. In: Banerjee, S., Rondoni, L. (eds) Applications of Chaos and Nonlinear Dynamics in Science and Engineering - Vol. 4. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-17037-4_7
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