عنوان مقاله [English]
نویسندگان [English]چکیده [English]
Random-like behaviors in various natural phenomena led researchers to apply more accurate forecasting methods. While statistical models are more traditional to use for such complex behaviors, Chaos theory has been emerging as a new way to hydrologists and water engineers. In the Chaos theory viewpoint, random-like behavior can be related to a simple determinism which is hidden in the background of system dynamics and can be shown in an optimized phase space. If the conditions of a chaotic system which Chaos theory has been stated dominate the system behavior, dynamics in the phase space follows a fractal pattern which is aforementioned hidden determinism and called the attractor. The case study is the Kashkan River, which is located in Lorestan province, southwestern of Iran, where semi- arid climate is predominant. While the average number of dry days reach to 185 in a year, average precipitation is 375.3 mm during a water year. In the present study, the daily runoff time series of the Kashkan River have been analyzed using Chaos theory, following its observed random-like behavior. To perform chaotic analysis, a phase space should be reconstructed by determining optimize time delay and embedding dimension. So far, various methods have been suggested to calculate the time delay, including the Average Mutual Information method which have gained more popularity among the others. In this paper, a new method has been presented to estimate optimum time delay in the base of the AMI method. Therefore, instead of using the first local minimum of Mutual Information Function, its overall minimum has been considered to estimate optimum time delay. This method has been applied to daily runoff time series of the Kashkan River and its efficiency has been studied in fractal dimension estimation methods. While False Nearest Neighbors and Correlation Dimension methods have been employed to evaluate fractal dimension of system attractor, sensitivity to initial conditions have been studied using Lyapunov Exponent and Kolmogorov Entropy methods as the other major feature of chaotic behavior.
In the first part of this study, the AMI method and Mutual Information Function have been examined theoretically by evaluating its performance in a chaotic map called Rossler Map. It has been shown that the first minimum of MIF can be an optimum option to form best illustration of system chaotic attractor. While the first local minimum is more effective to mathematical functions such as RM, some other issues should be considered when the case study is a natural system, the MIF can have many minimums in different local sections. As mentioned earlier, the overall minimum of MIF have been used to determine the time delay as a new method phase space reconstruction which is obtained ? =107 days. False Nearest Neighbors and Correlation Dimension method have been used to evaluate fractal dimension, which showed the existence of a fractal attractor in phase space, and also the superiority of this new method for phase space reconstruction. Positive Lyapunov Exponent is a sign of sensitivity to initial conditions and therefore, chaotic behavior in system, which is more emphasized by calculating Kolmogorov Entropy in 3 different radius. Regardless of the selected radius, KE reaches to a finite number which is an evidence of chaotic behavior. While a certain number can not be calculated for Kolmogorov Entropy, it can be observed that by selecting a smaller radius, KE reaches to a finite positive number. Hence, the system entropy is consistent and follows a chaotic pattern.
Results indicated Chaos existence in this time series and suitability of Chaos theory-based models for governing system of flow in the Kashkan River. Sensitivity to initial conditions and fractal attractor as two major characteristics of chaotic system have been observed. In addition, a new AMI-based method to determine the optimum time delay has been showed better indication of system chaotic behavior in phase space. Compared with a traditional approach, the attractor which has been formed in reconstructed phase space behaves more constantly by increasing the embedding dimension in both FNN and CD methods. In comparison to similar studies analyzed the times series with different lengths, it can be concluded that the length of a time series cannot effectively adjust overall conclusion in the chaotic analysis of the Kashkan river flow governing system. Generally, evidences of chaotic behavior in the Kashkan river flow time series have been confirmed. Hence, employing chaotic models can be very helpful to forecast these time series.