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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.

One aspect which hydrologists have been extensively working on is the structure of hydrological processes, such as rainfall and runoff [

Against the backdrop of the dynamics governing river flow, therefore, investigating the existence of a chaotic component and more generally, nonlinear time sequence analysis may be useful in understanding some aspects of the phenomenon of river flow process. This is imperative considering the fact that river flow dynamics is linked both to the climate, through precipitations, radiation, etc., and to the inflow-runoff transformation. Thus, the existence of chaos cannot be excluded a priori. Generally, it should be noted that there are some aspects of the inflow-runoff transformation which may reduce the complexity of the local dynamics of the climate, shading some degrees of freedom and privileging others. For instance, especially, the geometric, hydraulic, and hydrogeological characteristics of the watershed [

In a simple contextual framework, the term “chaos” is used to denote the irregular behaviour of dynamical systems arising from a strictly deterministic time evolution without any source of external stochasticity but with sensitivity to initial conditions. From a physical point of view, the most striking difference between a picture based on a stochastic description and the approach based on deterministic chaos is essentially contained in the very different number of variables, which characterise the system [

Thus, determining the presence of low-dimensional attractor from time series data has important dynamical implications. Hence, the calculation of the system dimension is one of the steps toward this goal. The central idea behind the application of this approach is that systems whose dynamics are governed by stochastic processes are thought to have an infinite value for the attractor dimension, whereas a finite, low, non-integer value of the dimension is considered to be a strong indication of the presence of deterministic chaos [

In this study, historical time series for gauging stations at the base of the Benue River (i.e., Lower Benue River Basin) at Makurdi (7˚44′N, 8˚32′E) was used. A total of 26 years (1974-2000) water stage and daily discharge data were collected. The Benue River is the major tributary of the Niger River. It is approximately 1400 km long and almost navigable during the rainy season (between July and October). Its headwaters rise in the Adamawa Plateau of the Northern Cameroon, flows into Nigeria south of the Mandara Mountains through the east-central part of Nigeria. There is only one high-water season because of its southerly location; this normally occurs from May to October, while on the other hand, the low-water period is from December to June.

hydrological flow regime of the Benue River in line with the general climatic pattern. There are definite wet and dry seasons which give rise to changes in river flow and salinity regimes. The flood of the Benue River (upper, middle, and downstream) lasts from July to October, and sometimes up to early November.

The mean daily discharges are as shown in

The first step in the search for a deterministic behaviour is that of attempting to reconstruct the dynamics in phase space. Having available the time series of only one of the variables present in the phenomenon, that is, the discharge, the delay time method proposed by Takens [

where, is called the embedding dimension.

To construct a well-behaved phase space by delay time, a careful choice of is critical. The delay time is commonly selected by using the autocorrelation

function (ACF) method where ACF first attains zeros or drops below a small value, say, or the mutual information (MI) method according to Fraser and Swinney [

The most commonly used algorithm for computing the correlation dimension is the Grassberger-Procaccia algorithm [

subject to the constraints:

where, is the Heaviside step function, and r is the radius between the pair of points in phase space. The straightforward estimator, Equation (2), is biased towards too small dimensions when the pairs entering the sum are statistically independent [

In order to avoid the problem of temporal correlation, a modified form of Equation (2) was used in the computation of the correlation dimension, as given by Equation (3) and implemented in the TISEAN 3.0 Software package.

where, n_{min} is a threshold value such that pairs of vectors in the m-dimensional phase space which are closer in time than it are discarded to avoid temporal correlations that may contaminate the result. For the implementation of Equation (3), n_{min} was set to 182 for the daily flow series as suggested by Wang et al., [

Since a positive maximal Lyapunov exponent is a strong signature of deterministic chaos, it is of considerable interest to determine its value; at least to complement the results obtained with the correlation dimension method. The first algorithm for this purpose was suggested by Wolf et al., [

The use of chaotic models to forecast follows the recent trend of viewing most of the natural phenomena in a nonlinear way. In the chaotic approach, the governing assumption is that the discharges are driven by a multivariate dynamical system and observed through the equation . Let m be the dimension of the attractor of the original and unobservable system ; the characteristics of the dynamics can be analyzed in a reconstructed phase space obtained by the d-dimensional vectors , where is called delay time and d, embedding dimension. As a consequence, in the reconstructed phase space there exists a deterministic dynamic f such that

Equation (5) can be used to forecast the future of the time series and a good prediction depends, essentially, on the ability of approximating the dynamic f with an estimate f. To execute nonlinear prediction as described above, there is the need to find an approximation of f in Equation (5). This can be done by several methods, starting with polynomial representations, from Kernel or Spline methods, up to Neural Networks, Wavelets or the Nearest Neighbours’ method. The nonlinear prediction was performed by adopting the method of local constant approximation using the Fast Neighbour search algorithm [

Analyses of correlation dimension for the presence of deterministic chaos in streamflow series for timescales of daily, monthly and discharge derivative values show contrasting results. For instance, Savard [

The essence of the discussion above is to put the existing results in perspective so as to provide an objective framework for drawing conclusions on the findings in this study. It is important to note that some authors do not provide scaling region when reporting on the investigation aimed at determining the existence of chaos (e.g., [11,16,18]), whereas others do, but give no obvious scaling region (e.g., [

Looking at the result obtained further, given the limitations of the autocorrelation function approach, which was used here, an effort was made to study the effect of the delay time on the correlation dimension estimate.

For a time series generated by deterministic dynamical systems, positive characteristic exponents indicate the presence of chaos; this is why it is sufficient to calculate the largest Lyapunov exponent (λ).

values of the Lyapunov exponent computed for the daily streamflow series. The values of λ reasonably are constant over a range of m between 3 and 28; based on the recommendation of Jayawardena and Lai [

As reported in Casdagli [

The study of the dynamics which controls the evolution of river flow, conducted in the light of chaos theory may have conflicting results. Some of these are more of a speculative character whereas others may have practical potentials. Following from the nonlinear analyses in this study, there is no discernible evidence to suggest the existence of low-dimensional chaos in the daily streamflow process of the Benue River. Though contradictory results abound on the existence of low-dimensional chaos in daily streamflows, it is paramount to note that

since the dynamics of observed hydrological series is inevitably contaminated by not only measurement noise, but also dynamical noise, one may contend that the daily streamflow time series may span a wide dynamical range between deterministic chaos and periodic signal contaminated with additive noise. Added noise (measurement error) may strongly affect the nonlinear behaviour of deterministic system by decreasing the predictability, and invariably increases the dimension of an existing attractor. Thus, because streamflow process usually suffers from strong natural and anthropogenic disturbances which are themselves made up of both stochastic and deterministic components, it does beg the question of how a reliable identification of chaotic dynamics can be made even if the streamflow is actually low-dimensional. Considering also, for instance, the correlation dimension analysis provides information only on the number of variables influencing the dynamics of the system and does not in any plausible way identify the variables or the mathematical model for the dynamics of the river flow process. Hence, evidence of the existence of chaotic characteristics does not in any physical way, translate directly to determinism. However, regardless of the contradictory reports on the result of findings on deterministic chaos, this concept provides a profound technique for time series analysis and thus invariably allows for an intuitive understanding of dynamical systems.