Application of Chaos Theory in Modeling and Analysis of River Discharge under Different Time Scales (Case Study: Karun River)

Introduction
One of the main issues in hydrology and water resources is investigation of river flow. Due to innovations and capabilities of the chaos theory, nowadays, chaos analyses are used to analyze river-flow time series. Since investigation of the presence of different characteristics at different time scales in rivers is one of the main challenges of hydrology in recent years, the aim of this paper is to study the behavior of river flow at different time scales. The behavior of river discharge can be studied precisely by applying nonlinear and chaotic analyses. The chaos theory, as the foundation of nonlinear dynamic systems has created great changes in understanding and expressing the mode of different phenomena in recent decades. This theory deals with the study of systems that at first glance may seem irregular; but in fact they are governed by clear rules. Such systems are very sensitive to primary conditions, so that seemingly minor inputs could have a significant impact on that. Such systems are called chaotic. With regard to recent studies, based on chaos theory for flow discharges, the chaotic or random nature of a system could be identified by using some discriminative indices. Despite chaotic studies conducted on the river discharges, chaotic analysis of flow discharge in Karun River has not been implemented for different time scales.
Materials and methods
In this study, the presence of chaos at daily, monthly and seasonal scales in discharge data of Karun River, Mollasani station, is discussed. Mollasani station is located downstream Ghir barrage (where, Dez, Gargar and Shotait River join together) and upstream Mollasani city. Daily, monthly and seasonal flow discharge data in Mollasani station (1967 to 2011) are used. Four nonlinear dynamic methods were used: 1) phase space reconstruction, 2) correlation dimension method, 3) largest Lyapunov exponent, and 4) spectral power. The state (phase) space is a useful tool for studying dynamic systems. According to this concept, a dynamic system can be described by means of a state space diagram. Each dynamic system consists of differential equations with partial derivatives. To determine these equations and their type, the embedding dimension and time delay parameter must be determined. The delay time could be obtained from the method of assessment of correlation function (ACF) or average mutual information (AMI). In this study, the average mutual information is used to estimate delay time of the dynamic system. In this method, time of first minimum occurrence in the average mutual information function is selected as the appropriate delay time. The embedding dimension is obtained from the false nearest neighbor (FNN) method. This algorithm provided information concerning optimal embedding dimension for the dynamic system.
Results and discussion
 The results showed that the daily times for daily, monthly and seasonal data are 97, 2 and 1, respectively, and the optimal embedding dimensions are 9, 6 and 2, respectively. To determine chaotic nature of the system, correlation coefficient was calculated. The correlation dimension at the monthly scales, due to saturation of the diagram, is obtained as 2.704. Therefore, Karun River system is chaotic at this scale. But at the daily and seasonal scales, the diagram's trend was ascending and as a result, the river discharge is random. Another indicative criterion of the chaotic system is the largest Lyapunov exponent. The behavior could be measured in each dimension by using the Lyapunov exponent. Presence of positive Lyapunov exponent is an important indicator of the chaotic system. In this study, elongation factors and largest Lyapunov exponent are calculated on the basis of Rosenstein's method. Taking the value of optimal embedding dimension as m, the value of this exponent can be calculated. In the absence of the optimal embedding dimension, this parameter is predicted based on different m values. At monthly tile scale, the largest Lyapunov exponent was positive (0.0093). The extent of band width at monthly scale is another proof of chaotic nature of this river's discharge. The chaotic nature of the discharge data can also be calculated by power range. These methods can estimate the chaotic or non-chaotic behavior and cannot estimate the complexity of data.
Conclusion
At daily and seasonal scales, according to correlation dimension, the river discharge is random (non-chaotic). But, flow is chaotic at the monthly scale. It seems that the geographical location of Mollasani station may affect the chaotic or randomness of Karun River's discharge.