Abstract
Complex systems such as river flow do not obey the law of linearity but most often than not behave in a nonlinear manner. In this study, the dynamics of the River Niger discharge along two stations located in Nigeria were investigated using the methods of phase space reconstruction, correlation dimension and Lyapunov exponent. The analysis was carried out by using the daily data for three different periods — 1914–1939 (period 1), 1940–1964 (period 2), and 1966–1991 (period 3) spanning before and after the Kainji dam construction. Time delay embedding values at Baro and Lokoja for the three time periods were used to reconstruct the phase space of the river discharge at both locations. The embedding dimension represents the number of variables to completely describe the system. The presence of chaos was confirmed with positive values of Lyapunov exponents in both locations at the three different periods considered. The Lyapunov exponents at Baro were in the range 0.0014–0.0150 while the range at Lokoja was 0.007–0.0145. Significant correlation dimension values obtained at Baro (2.11–2.81) and Lokoja (2.28–4.51) are indicative of low dimensional chaos. A possible explanation for this could be the temporal sequence in the dam installation. The forecast horizon, the inverse of the largest Lyapunov exponents, gives a prediction boundary on a chaotic time series. In this study, the forecast horizon for both locations is expected to be in the range of 40–58 days into the future.
Similar content being viewed by others
References
Adamson PT, Rutherfurd ID, Peel MC, Conlan IA (2009) The hydrology of the mekong river. In: The Mekong. Elsevier, pp 53–76
Adelekan IO (1998) Spatio-temporal variations in thunderstorm rainfall over Nigeria. Int J Climatol: J R Meteorol Soc 18(11):1273–1284
Adeyemi O, Fashae OA (2018) Urbanisation and hydraulic geometry response: a model approach. Int J Water 12(2):103–115
Alami M, Ghorbani M, Malekani L (2013) Chaotic analysis and prediction of river flows. J Hydraul Struct 1(1):54–64
Alfaro M, Fuertes G, Vargas M, Sepúlveda J, Veloso-Poblete M (2018) Forecast of chaotic series in a horizon superior to the inverse of the maximum Lyapunov exponent. Complexity, 2018
Alimohammadi H, Dastjerdi KY, Yaghin ML (2019) The study of progressive collapse in dual systems. Eng Arch 16(1):1–10
Babovic V, Keijzer M (2000) Forecasting of river discharges in the presence of chaos and noise. In: Flood issues in contemporary water management. Springer, pp 405–419
Beauvais A, Dubois J (1995) Attractor properties of a river discharge dynamical system. Eos Transations AGU 73(46):F234
Boon MY, Henry BI, Suttle CM, Dain SJ (2008) The correlation dimension: a useful objective measure of the transient visual evoked potential? J Vis 8(1):6–6
Fashae O, Olusola A, Adedeji O (2017) Geospatial analysis of changes in vegetation cover over Nigeria. Bull Geogr Phys Geogr Ser 13(1):17–27
Fashae O, Olusola A, Faniran A (2018) Environmental factors and pattern of riparian vegetation along the downstream sections of the Lower Ogun River, Nigeria. Singap J Trop Geogr 39(2):215–223
Fraedrich K (1986) Estimating the dimensions of weather and climate attractors. J Atmos Sci 43(5):419–432
Fraser AM, Swinney HL (1986) Independent coordinates for strange attractors from mutual information. Phys Rev A 33(2):1134
Fuwape IA, Ogunjo ST (2013) Investigating chaos in the Nigerian asset and resource management (arm) discovery fund. CBN J Appl Stat 4(2):129–140
Fuwape IA, Ogunjo ST, Oluyamo S, Rabiu A (2017) Spatial variation of deterministic chaos in mean daily temperature and rainfall over Nigeria. Theor Appl Climatol 130(1):119–132
Ghorbani MA, Kisi O, Aalinezhad M (2010) A probe into the chaotic nature of daily streamflow time series by correlation dimension and largest Lyapunov methods. Appl Math Model 34(12):4050–4057. https://doi.org/10.1016/j.apm.2010.03.036, http://www.sciencedirect.com/science/article/pii/S0307904X10001575
Graf WL (2006) Downstream hydrologic and geomorphic effects of large dams on American rivers. Geomorphology 79(3-4):336– 360
Grassberger P, Procaccia I (1983) Characterization of strange attractors. Phys Rev Lett 50 (5):346
Hohensinner S, Hauer C, Muhar S (2018) River morphology, channelization, and habitat restoration. In: Riverine ecosystem management. Springer, Cham, pp 41–65
Jayawardena A, Lai F (1994) Analysis and prediction of chaos in rainfall and stream flow time series. J Hydrol 153(1-4):23–52
Kędra M (2014) Deterministic chaotic dynamics of Raba river flow (Polish carpathian mountains). J Hydrol 509:474–503
Kennel MB, Brown R, Abarbanel HD (1992) Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys Rev A 45(6):3403
Khatibi R, Sivakumar B, Ghorbani MA, Kisi O, Koa̧k K, Zadeh DF (2012) Investigating chaos in river stage and discharge time series. J Hydrol 414:108–117
Koutsoyiannis D (2006) On the quest for chaotic attractors in hydrological processes. Hydrol Sci J 51(6):1065–1091
Latrubesse EM, Stevaux JC, Sinha R (2005) Tropical rivers. Geomorphology 70(3-4):187–206
Leopold LB, Wolman MG, Miller JP (1995) Fluvial processes in geomorphology. Courier Corporation
Magilligan FJ, Nislow KH (2005) Changes in hydrologic regime by dams. Geomorphology 71 (1–2):61–78
Mallen-Cooper M, Zampatti BP (2018) History, hydrology and hydraulics: rethinking the ecological management of large rivers. Ecohydrology 11(5):e1965
Mihailovic DT, Nikolicoric E, Arsenic I, Malinovic Milicevic S, Singh VP, Stosic T, Stosic B (2019) Analysis of daily streamflow complexity by Kolmogorov measures and Lyapunov exponent. Physica A: Stat Mech Applic 525:290–303
Mitschke F, Dammig M (1993) Chaos versus noise in experimental data. Int J Bifur Chaos 3 (03):693–702
Ng W, Panu U, Lennox W (2007) Chaos based analytical techniques for daily extreme hydrological observations. J Hydrol 342(1–2):17–41
Ogunjo S, Fuwape I, Oluyamo S, Rabiu B (2019) Spatial dynamical complexity of precipitation and temperature extremes over Africa and South America. Asia-Pac J Atmos Sci, 1–14
Ogunjo S, Rabiu A, Fuwape I, Obafaye A (2021) Evolution of dynamical complexities in geospace as captured by dst over four solar cycles 1964–2008. J Geophys Res: Space Phys 126(4):e2020JA027873
Ogunjo ST, Fuwape IA (2020) Nonlinear characterization and interaction in teleconnection patterns. Adv Space Res 65(12):2723– 2732
Olusola A (2019) Process-form dynamics of upper Ogun river basin, southwestern Nigeria. Unpublished PhD thesis submitted to the department of geography University of Ibadan
Olusola AO, Fashae O (2017) Stream energy distribution below Eleyele dam in Southwestern Nigeria. Singap J Trop Geogr 38(3):402–413
Oyebande L (1995) Effects of reservoir operation on the hydrological regime and water availability in northern Nigeria. IAHS Publications-Series of Proceedings and Reports-Intern Assoc Hydrological Sciences 230 (1995):25–34
Pasternack GB (1999) Does the river run wild? assessing chaos in hydrological systems. Adv Water Resour 23(3):253–260
Phillips JD (1992a) Deterministic chaos in surface runoff. Overland Flow: Hydraulic and Erosion Mechanics, pp 177–197
Phillips JD (1992b) Nonlinear dynamical systems in geomorphology: revolution or evolution? Geomorphology 5(3–5):219–229
Potter PE (1978) Significance and origin of big rivers. J Geol 86(1):13–33
Reiterer P, Lainscsek C, Schürrer F, Letellier C, Maquet J (1998) A nine-dimensional lorenz system to study high-dimensional chaos. J Phys A Math Gen 31(34):7121
Rosenstein MT, Collins JJ, De Luca CJ (1993) A practical method for calculating largest Lyapunov exponents from small data sets. Physica D: Nonlinear Phenomena 65(1-2):117–134
Savard C (1990) Correlation integral analysis of South Twin River streamflow, central nevada: preliminary application of chaos theory. Eos Transaction AGU 71(43):1341
Schölzel C (2019) Nonlinear measures for dynamical systems. Zenodo 0.5.2, https://doi.org/10.5281/zenodo.3814723
Schreiber T (1993) Extremely simple nonlinear noise-reduction method. Phys Rev E 47(4):2401
Sivakumar B (2000) Chaos theory in hydrology: important issues and interpretations. J Hydrol 227(1–4):1–20
Sivakumar B (2016) Chaos in hydrology: bridging determinism and stochasticity. Springer
Sivakumar B (2017) Applications to river flow data. In: Chaos in hydrology. Springer, pp 267–295
Srivalli CN, Jothiprakash V, Sivakumar B (2019) Complexity of streamflows in the west-flowing rivers of India. Stoch Env Res Risk A 33(3):837–853
Stevaux JC, Martins DP, Meurer M (2009) Changes in a large regulated tropical river: The Paraná river downstream from the Porto Primavera dam, Brazil. Geomorphology 113(3-4):230–238
Stosic T, Telesca L, de Souza Ferreira DV, Stosic B (2016) Investigating anthropically induced effects in streamflow dynamics by using permutation entropy and statistical complexity analysis: a case study. J Hydrol 540:1136–1145
Takens F (1981) Detecting strange attractors in turbulence. In: Dynamical systems and turbulence, Warwick 1980. Springer, pp 366–381
Tongal H, Sivakumar B (2019) Entropy analysis for spatiotemporal variability of seasonal, low, and high streamflows. Stoch Environ Res Risk Assessment 33(1):303–320
Wang X, Lei T (2017) Hydrologic system behavior characteristic analysis and long-term prediction based on chaos radial basis function networks. Boletín Técnico 55(17):536–546
Wohl EE (2007) Hydrology and discharge. Large rivers: Geomorphology and management, 29–44
Xiaofeng G, Lai CH (2000) Detecting chaos from time series. J Phys A: Math General 33 (5):1007–1016. https://doi.org/10.1088/0305-4470/33/5/313
Zadehmohamad M (2020) Evaluation of cyclic displacement amplitude of integral bridge abutment on backfill soil pressure and settlements. Journal of Transportation Research In press, 1–10
Zheng J, He H, Alimohammadi H (2021) Three-dimensional wadell roundness for particle angularity characterization of granular soils. Acta Geotech 16(1):133–149
Zhou Y, Ma Z, Wang L (2002) Chaotic dynamics of the flood series in the Huaihe River Basin for the last 500 years. J Hydrol 258(1–4):100–110
Zounemat-Kermani M (2016) Investigating chaos and nonlinear forecasting in short term and mid-term river discharge. Water Resour Manag 30(5):1851–1865
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Responsible Editor: Broder J. Merkel
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ogunjo, S., Olusola, A., Fuwape, I. et al. Temporal variation in deterministic chaos: the influence of Kainji dam on downstream stations along lower Niger River. Arab J Geosci 15, 237 (2022). https://doi.org/10.1007/s12517-021-09297-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12517-021-09297-0