Nonlinear Deterministic Chaos in Benue River Flow Daily Time Sequence

ABSTRACT

The Various physical mechanisms governing river flow dynamics act on a wide range of temporal and spatial scales. This spatio-temporal variability has been believed to be influenced by a large number of variables. In the light of this, an attempt was made in this paper to examine whether the daily flow sequence of the Benue River exhibits low-dimensional chaos; that is, if or not its dynamics could be explained by a small number of effective degrees of freedom. To this end, nonlinear analysis of the flow sequence was done by evaluating the correlation dimension based on phase space reconstruction and maximal Lyapunov estimation as well as nonlinear prediction. Results obtained in all instances considered indicate that there is no discernible evidence to suggest that the daily flow sequence of the Benue River exhibit nonlinear deterministic chaotic signatures. Thus, it may be conjectured that the daily flow time series span a wide dynamical range between deterministic chaos and periodic signal contaminated with additive noise; that is, by either measurement or dynamical noise. However, contradictory results abound on the existence of low-dimensional chaos in daily streamflows. Hence, it is paramount to note that if the existence of low-dimension deterministic component is reliably verified, it is necessary to investigate its origin, dependence on the space-time behavior of precipitation and therefore on climate and role of the inflow-runoff mechanism.

The Various physical mechanisms governing river flow dynamics act on a wide range of temporal and spatial scales. This spatio-temporal variability has been believed to be influenced by a large number of variables. In the light of this, an attempt was made in this paper to examine whether the daily flow sequence of the Benue River exhibits low-dimensional chaos; that is, if or not its dynamics could be explained by a small number of effective degrees of freedom. To this end, nonlinear analysis of the flow sequence was done by evaluating the correlation dimension based on phase space reconstruction and maximal Lyapunov estimation as well as nonlinear prediction. Results obtained in all instances considered indicate that there is no discernible evidence to suggest that the daily flow sequence of the Benue River exhibit nonlinear deterministic chaotic signatures. Thus, it may be conjectured that the daily flow time series span a wide dynamical range between deterministic chaos and periodic signal contaminated with additive noise; that is, by either measurement or dynamical noise. However, contradictory results abound on the existence of low-dimensional chaos in daily streamflows. Hence, it is paramount to note that if the existence of low-dimension deterministic component is reliably verified, it is necessary to investigate its origin, dependence on the space-time behavior of precipitation and therefore on climate and role of the inflow-runoff mechanism.

Cite this paper

nullO. Martins, M. Sadeeq and I. Ahaneku, "Nonlinear Deterministic Chaos in Benue River Flow Daily Time Sequence,"*Journal of Water Resource and Protection*, Vol. 3 No. 10, 2011, pp. 747-757. doi: 10.4236/jwarp.2011.310085.

nullO. Martins, M. Sadeeq and I. Ahaneku, "Nonlinear Deterministic Chaos in Benue River Flow Daily Time Sequence,"

References

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[11] S. Bellie, L. Shie-Yui, L. Chih-Young and P. Kok-Kwang, “Singapore Rainfall Behaviour: Chaotic?” Journal of Hydrologic Engineering, Vol. 4, No. 1, 1999, pp. 38-48. doi:10.1061/(ASCE)1084-0699(1999)4:1(38)

[12] F. Takens, “Detecting Strange Attractors in Turbulence,” In: D.A. Rand and L.S. Young, Eds., Lecture Notes in Mathematics, Vol. 898, Springer-Verlag, New York, 1981, pp. 366-381.

[13] N. H. Packard, J. P. Crutchfield, J. D. Farmer and R. S. Shaw, “Geometry from a Time Series,” Physical Review Letters, Vol. 45, No. 9, 1980, pp. 712-716. doi:10.1103/PhysRevLett.45.712

[14] A. Fraser and H. L. Swinney, “Independent Coordinates for Strange Attractors from Mutual Information,” Physical Review A, Vol. 33, No. 2, 1986, pp. 1134-1140. doi:10.1103/PhysRevA.33.1134

[15] G. J. Mpitsos, H. C. Creech, C. S. Cohan and M. Mendelson, “Variability and Chaos: Neuron-Integrative Principles in Self-Organization of Motor Patterns,” In: B.-L. Hao, Ed., Directions in Chaos, Vol. 1, 1987, World Scientific, pp. 162-190.

[16] A. W. Jayawardena and F. Lai, “Analysis and Prediction of Chaos in Rainfall and Streamflow Time Series,” Journal of Hydrology, Vol. 153, No. 1-4, 1994, pp. 23-52. doi:10.1016/0022-1694(94)90185-6

[17] M. N. Islam and B. Sivakumar, “Characterization and Prediction of Runoff Dynamics: A Nonlinear Dynamical View,” Advances in Water Resources, Vol. 25, No. 2, 2002, pp. 179-190. doi:10.1016/S0309-1708(01)00053-7

[18] A. Elshorbagy, S. P. Simonovic and U. S. Panu, “Estimation of Missing Streamflow Data Using Principles of Chaos Theory,” Journal of Hydrology, Vol. 255, 2002, pp. 125-133. doi:10.1016/S0022-1694(01)00513-3

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[20] W. Wang, P. H. A. J. M. Van Gelder and J. K. Vrijing; “Detection of Changes in Streamflow Series in Western Europe over 1901-2000,” Water Science and Technology, Water Supply, Vol. 5, No. 6, 2005, pp. 289-299.

[21] G. B. Pasternack, “Does the River Run Wild? Assessing Chaos in Hydrological Systems,” Advances in Water Resources, Vol. 23, No. 3, 1999, pp. 253-260. doi:10.1016/S0309-1708(99)00008-1

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[26] H. Kantz, “A Robust Method to Estimate the Maximal Lyapunov Exponent of a Time Series,” Physical Letters A, Vol. 185, No. 1, 1994, pp. 77-87. doi:10.1016/0375-9601(94)90991-1

[27] C. S. Savard, “Looking for Chaos in Streamflow with Discharge Derivative Data,” EOS Trans AGU (Spring Meeting suppl.), Vol. 73, No. 14, 1992, p. 50

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[30] L. Francesco and V. Villi, “Chaotic Forecasting of Discharge Time Series: A Case Study,” Journal of the American Water Resources Association, Vol. 87, No. 2, 2001, pp. 271-279.

[31] A. Provenzale, A. Smith, R. Vio and G. Murante, “Distinguishing between Low Dimensional Dynamics and Randomness in Measured Time Series,” Physica D, Vol. 58, No. 1, 1992, pp. 31-49. doi:10.1016/0167-2789(92)90100-2

[32] G. Kember and A. C. Flower, “Forecasting River Flow Using Nonlinear Dynamics,” Stochastic Hydrology and Hydraulics, Vol. 7, 1993, pp. 205-212. doi:10.1007/BF01585599

[33] M. Casdagli, “Nonlinear Prediction of Chaotic Time Series,” Physica D, Vol. 35, 1989, pp. 335-356. doi:10.1016/0167-2789(89)90074-2

[34] G. Sugihara and R. M. May, “Nonlinear Forecasting as a Way of Distinguishing Chaos from Measurement Error in Time Series,” Nature, Vol. 344, 1990, pp. 734-741. doi:10.1038/344734a0

[1] B. Sivakumar, “Chaos Theory in Hydrology: Important Issues and Interpretations,” Journal of Hydrology, Vol. 227, No. 1-4, 2000, pp. 1-20. doi:10.1016/S0022-1694(99)00186-9

[2] P. Amilcare and L. Ridolfi, “Nonlinear Analysis of River Flow Time Series,” Water Resources Research, Vol. 33, No. 6, 1997, pp. 1353-1367. doi:10.1029/96WR03535

[3] P. Grassberger, “An Optimized Box-Assisted Algorithm for Fractal Dimensions,” Physical Review Letters A, Vol. 148, 1991, pp. 521-522.

[4] H. Inaoka and H. Takayasu, “Water Erosion as a Fractal Growth Process,” Physical Review E, Vol. 47, No. 2, 1993, pp. 899-910. doi:10.1103/PhysRevE.47.899

[5] S. D. Peckman, “New Results for Self-Similar Trees with Applications to River Networks,” Water Resources Research, Vol. 31, No. 4, 1995, pp. 1023-1029. doi:10.1029/94WR03155

[6] A. Rinaldo, G. K. Vogel, R. Rigon and I. Rodriguez- Iturbe, “Can One Gauge the Shape of a Basin?” Water Resources Research, Vol. 31, No. 4, 1995, pp. 1119- 1127. doi:10.1029/94WR03290

[7] C. Nicolis and G. Nicolis, “Is There a Climatic Attractor?” Nature, Vol. 311, 1984, pp. 529-532. doi:10.1038/311529a0

[8] K. Fraedrich, “Estimating the Dimensions of Weather and Climate Attractors,” Journal of the Atmospheric Sciences, Vol. 43, No. 5, 1986, pp. 419-432. doi:10.1175/1520-0469(1986)043<0419:ETDOWA>2.0.CO;2

[9] K. Fraedrich, “Estimating Weather and Climate Predictability on Attractors,” Journal of the Atmospheric Sciences, Vol. 44, No. 4, 1987, pp. 722-728. doi:10.1175/1520-0469(1987)044<0722:EWACPO>2.0.CO;2

[10] P. Grassberger, “Do Climatic Attractors Exist?” Nature, Vol. 323, 1986, pp. 609-612. doi:10.1038/323609a0

[11] S. Bellie, L. Shie-Yui, L. Chih-Young and P. Kok-Kwang, “Singapore Rainfall Behaviour: Chaotic?” Journal of Hydrologic Engineering, Vol. 4, No. 1, 1999, pp. 38-48. doi:10.1061/(ASCE)1084-0699(1999)4:1(38)

[12] F. Takens, “Detecting Strange Attractors in Turbulence,” In: D.A. Rand and L.S. Young, Eds., Lecture Notes in Mathematics, Vol. 898, Springer-Verlag, New York, 1981, pp. 366-381.

[13] N. H. Packard, J. P. Crutchfield, J. D. Farmer and R. S. Shaw, “Geometry from a Time Series,” Physical Review Letters, Vol. 45, No. 9, 1980, pp. 712-716. doi:10.1103/PhysRevLett.45.712

[14] A. Fraser and H. L. Swinney, “Independent Coordinates for Strange Attractors from Mutual Information,” Physical Review A, Vol. 33, No. 2, 1986, pp. 1134-1140. doi:10.1103/PhysRevA.33.1134

[15] G. J. Mpitsos, H. C. Creech, C. S. Cohan and M. Mendelson, “Variability and Chaos: Neuron-Integrative Principles in Self-Organization of Motor Patterns,” In: B.-L. Hao, Ed., Directions in Chaos, Vol. 1, 1987, World Scientific, pp. 162-190.

[16] A. W. Jayawardena and F. Lai, “Analysis and Prediction of Chaos in Rainfall and Streamflow Time Series,” Journal of Hydrology, Vol. 153, No. 1-4, 1994, pp. 23-52. doi:10.1016/0022-1694(94)90185-6

[17] M. N. Islam and B. Sivakumar, “Characterization and Prediction of Runoff Dynamics: A Nonlinear Dynamical View,” Advances in Water Resources, Vol. 25, No. 2, 2002, pp. 179-190. doi:10.1016/S0309-1708(01)00053-7

[18] A. Elshorbagy, S. P. Simonovic and U. S. Panu, “Estimation of Missing Streamflow Data Using Principles of Chaos Theory,” Journal of Hydrology, Vol. 255, 2002, pp. 125-133. doi:10.1016/S0022-1694(01)00513-3

[19] B. P. Wilcox, M. S. Seyfried, T. H. Matison, “Searching for Chaotic Dynamics in Snowmelt Runoff,” Water Resources Research, Vol. 27, No. 6, 1991, pp. 1005-1010.

[20] W. Wang, P. H. A. J. M. Van Gelder and J. K. Vrijing; “Detection of Changes in Streamflow Series in Western Europe over 1901-2000,” Water Science and Technology, Water Supply, Vol. 5, No. 6, 2005, pp. 289-299.

[21] G. B. Pasternack, “Does the River Run Wild? Assessing Chaos in Hydrological Systems,” Advances in Water Resources, Vol. 23, No. 3, 1999, pp. 253-260. doi:10.1016/S0309-1708(99)00008-1

[22] P. Grassberger and I. Procaccia, “Measuring the Strangeness of Strange Attractors,” Physica D, Vol. 9, No. 1-2, 1983, pp. 189-208. doi:10.1016/0167-2789(83)90298-1

[23] K. Holger and T. Schreiber, “Nonlinear Time Series Analysis,” Cambridge University Press, Cambridge, 1997, pp. 42-86.

[24] A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, “Determining Lyapunov Exponents from a Time Series,” Physica D, Vol. 16, No. 3, 1985, pp. 285-317. doi:10.1016/0167-2789(85)90011-9

[25] M. T. Rosenstein, J. J. Collins and C. J. De Luca, “A Practical Method for Calculating Largest Lyapunov Exponents for Small Data Sets,” Physica D, Vol. 65, 1993, pp. 117-134. doi:10.1016/0167-2789(93)90009-P

[26] H. Kantz, “A Robust Method to Estimate the Maximal Lyapunov Exponent of a Time Series,” Physical Letters A, Vol. 185, No. 1, 1994, pp. 77-87. doi:10.1016/0375-9601(94)90991-1

[27] C. S. Savard, “Looking for Chaos in Streamflow with Discharge Derivative Data,” EOS Trans AGU (Spring Meeting suppl.), Vol. 73, No. 14, 1992, p. 50

[28] E. Ott, “Chaos in Dynamical Systems,” Cambridge University Press, New York, 1993.

[29] J. W. Havstad and G. Mayer-Kress, “Attractor Dimension of Non-Stationary Dynamical Systems from Small Data Sets,” Physical Review A, Vol. 39, No. 2, 1989, pp. 845- 853. doi:10.1103/PhysRevA.39.845

[30] L. Francesco and V. Villi, “Chaotic Forecasting of Discharge Time Series: A Case Study,” Journal of the American Water Resources Association, Vol. 87, No. 2, 2001, pp. 271-279.

[31] A. Provenzale, A. Smith, R. Vio and G. Murante, “Distinguishing between Low Dimensional Dynamics and Randomness in Measured Time Series,” Physica D, Vol. 58, No. 1, 1992, pp. 31-49. doi:10.1016/0167-2789(92)90100-2

[32] G. Kember and A. C. Flower, “Forecasting River Flow Using Nonlinear Dynamics,” Stochastic Hydrology and Hydraulics, Vol. 7, 1993, pp. 205-212. doi:10.1007/BF01585599

[33] M. Casdagli, “Nonlinear Prediction of Chaotic Time Series,” Physica D, Vol. 35, 1989, pp. 335-356. doi:10.1016/0167-2789(89)90074-2

[34] G. Sugihara and R. M. May, “Nonlinear Forecasting as a Way of Distinguishing Chaos from Measurement Error in Time Series,” Nature, Vol. 344, 1990, pp. 734-741. doi:10.1038/344734a0