جستجوی مقالات مرتبط با کلیدواژه « Chaos theory » در نشریات گروه « عمران »

In this research prediction methods of artificial neural network and chaos theory are employed to predict daily, weekly and monthly runoff. For this, runoff series data observed at Pole-Kohneh located in the Qareh-Soo River. The nonlinear predictions of chaos are found to be in close agreement with the observed runoff, with high correlation coefficient for daily and weekly time scales. Predicted results of monthly time scale are not satisfying which indicating the chaos behavior in daily and weekly scales. The predicted results of ANN are inferior to chaos for daily and weekly scales but superior to chaos for monthly scale.

Unmanned aerial systems (UAS) are one of the latest technologies utilized in the hazard management and remote sensing. Nowadays, tendency in the development of UAS is toward autonomous navigation or hybrid tasks. In this context, development of comprehensive, efficient methodologies for path planning, control and navigation of UAS can be regarded as one of the fundamental steps for the development of autonomous systems. Up to now, different planning algorithms have been proposed in the specialized literature in order to enrich the framework of autonomous navigation of unmanned aerial systems. However, few efforts have been devoted to design new chaotic path planners for determining the optimal trajectories of these aerial systems in urban areas. An effective path planning technique can attain mission aims with respect to various restrictions of the UAS and less computational time.Chaos theory is one of the most studied theories with different applications in engineering and technology. Most of the natural processes demonstrate chaotic behavior such as black hole and clouds. Past researchers showed that if an evolutionary algorithm be hybridized with chaos, its performance will have improved, considerably. However, most of the evolutionary algorithms are inspired from nature, but all of their steps are random based motions. But nature is not either completely random based or chaotic. Hence, the combination of these theories should be more realistic. With this regard, evolution and chaos are related to each other narrowly in most of the complex natural systems. It is evidenced that some of the chaotic signals can alleviate the premature convergence problem of the evolutionary algorithms in tackling optimization problems.In this article, first, UAS path planning is modeled as a 3D constrained optimization problem. In this modeling, the aim is the optimization of path, fuel and safety with respect to different restrictions. After scheming and suggesting of general planning framework, UAS path planning problem is investigated by comparative study with regard to the studied scenario. For this aim, evolutionary planner is implemented in order to minimize the flight height, path length and energy consumption considering different restrictions such as safe altitude, turning angle, climbing slope, gliding slope, no fly zones and mission map limits. Then, a comprehensive model is employed to describe route-planning task, and then, based on the hybridization of chaos theory with evolutionary computing, four new evolutionary optimizers are developed. Hence, this paper developed four chaotic optimizers including particle swarm optimization, differential evolution, imperialist competitive algorithm and artificial bee colony technique based on 14 chaotic signals.In the rest of this paper, analyses, and extensive performance evaluation of the designed trajectory-planning approaches are performed according to the success rate results, precision and quality of the results, CPU running times, and convergence speed. The results show that the proposed framework can be utilized in represented scenario as an effective path planner. Proposed strategies are capable to compute the optimal paths more efficiently in comparison with the standard algorithms. From the results it is known that the chaotic differential evolution with logistic map can outperform the other compared algorithms.

Generally، The dynamics which is observed in time series of a hydrologic system variable have been considered as complex and random behavior. During last decades، using various artificial intelligence approaches such as chaos theory to analyze and prediction of hydrologic systems have been increased. In chaos theory viewpoint، complexity and random-like behavior of a system can be resulted from a simple and hidden determinism. Therefore، systems such as dominant hydrologic system which controls flow in a river can have this kind of determinism. If such determinism is existed، can be observed through system phase space، which can be reconstructed using a time series by lags method. Based on such a pattern that formed in reconstructed phase space، various prediction models can be used to forecast system behavior in future. Hence، chaotic behavior of the Kashkan river daily discharge time series have been studied using False Nearest Neighbors and Lyapunov Exponent methods which evaluated fractal attractor and sensitivity to initial condition as two major characteristics of a chaotic system. Average Mutual Information method was used to determine optimal delay time in phase space reconstruction by delay method. In this paper، it has been suggested to use first global minimum of mutual information function as standard to select optimal delay time. According to the results which have been obtain by these methods، chaotic behavior in daily runoff time series of the Kashkan river have been observed. In False Nearest Neighbors method، the percent of false neighbors have been significantly decreased due to rising embedding dimension of phase space، which have been shown the existence of a fractal attractor in system phase space. In lyapunov exponent method، the sensitivity to initial condition has been evaluated through reconstructed phase space and positive lyapunov exponent has been obtained. Hence، chaos theory-based models can be used to forecast daily runoff in this system. Various local models were used to make prediction based on reconstructed phase space and the results have been compared. Local Average and Local Polynomial was among local models that employed in this study. In addition، as a new hybrid approach، Multi Layer Prespetron Artificial Neural Networks have been used to local modeling based on phase space. All prediction results show appropriate quality of local prediction models in base of attractor pattern in phase space of dominant system of the Kashkan river flow. The accuracy which have been resulted from local hybrid model with Artificial Neural Networks، have been not shown significant difference with other current local models such as Local Average and Local Polynomial prediction methods. However، the Local Polynomial model has been shown better forecasting accuracy in compare with other methods. Totally، Local chaotic methods are suggested to make daily prediction of runoff in the Kashkan river.

In this paper, flow of the Kashkan River was analyzed through chaotic viewpoint regarding the daily discharge time series. At first, time series noise level was evaluated by the Gaussian Kernel estimation and wavelet transform methods. In addition, statistical behavior of time series was studied using autocorrelation and partial autocorrelation functions. Then, in phase space reconstruction by lags method, Average Mutual Information and False Nearest Neighbors methods were used to recognize the optimal delay time and embedding dimension, respectively. Subsequently, fractal dimension of system had been estimated using the correlation dimension method. In addition, sensitivity to initial conditions examined by Lyapunov exponent method and finally, prediction has been made using the local approximation method. Decrease of false neighbors due to increasing the embedding dimension, shows the existence of fractal attractor in system's phase space, which beside positive Lyapunov exponent obtained, suggests the condition of a chaotic system for river flow in the Kashkan basin. Following these results, forecasting had been made using local approximation method in the base of reconstructed phase space and satisfactory accuracy obtained, indicate the usefulness of chaos theorybased methods to analysis and prediction of river flow in the Kashkan basin. This efficiency was emphasized using comparison between local approximation and genetic programming results.

Significant wave height is mean of one third of the largest wave heights in a certain marine condition. Investigation and prediction of the significant wave height have been recently considered in marine system analysis including loadings over marine structures and sediment transport for designing, operation and marine researches. The capability of chaos theory in engineering particularly marine engineering has been gaining considerable interest in recent times. In this research, dynamic characteristics of the significant wave height time series in Caspian Sea at Anzali entrance are considered and the prediction has been performed using ideas gained from chaos theory. To reconstruct phase space, the time delay and embedding dimension are needed and for this purpose, autocorrelation function and algorithm of false nearest neighbors are used. Correlation dimension method is applied for investigating chaotic behavior of the significant wave height, which is the resultant of correlation dimensions, expresses chaotic behavior in the time series. Local prediction algorithm is used for time series prediction and results illustrate good and acceptable accuracy of chaos theory in quantitative prediction of seas significant wave height.

1. IntroductionThe study of river flow is one of the most important cases in the designing of a water storage structure and management of extreme events such as floods and droughts.The rate of river flow depends on various parameters and the nonlinear relationship between them has caused river behavior to be dynamic, nonlinear and complex.Chaos theory is the study of complex systems that, at first glance, do not appear to follow the regular laws of science. Chaos theory is one of the most fascinating and promising developments in the late 20th century mathematics and science. It provides a way of making sense out of phenomena such as river flow that seem to be totally without organization or order. A chaotic system is defined as a deterministic system in which small changes in the initial conditions may lead to completely different behavior in the future. Instability, non-periodic behavior, certain systems, being nonlinear, together is defined the chaotic systems. For the first time, chaos theory was used by Edward Lorenz in 1965 in meteorology. Later it has been implemented in all fields of science and empirical issues e.g. mathematics, behavioral, astronomy, mechanics, physics, mathematics, biology, economics etc.To date, a lot of attention has been devoted on analysing hydrological processes and elements by means of deterministic chaos approach. For example, Domenico and Ghorbani [1], Ghorbani et al. [2], Islam and Sivakumar [3], Sivakumar et al. [4] and Regonda et al. [5] have used nonlinear deterministic approaches to detect the presence of chaos and achieve more accurate river flow predictions.These investigations suggest that characterization (chaotic or stochastic) of river flow should be a necessary first step in any study, as it could provide important information on appropriate approaches for transforming data.There are other applications of chaos theory in the various topics that are not discussed in here. In this paper, the behaviour of river flow is forecasted by means of chaos theory.2. Methodology2.1. Case studyNahandchai drainage basin is located between the coordinates 38° 13' to 38° 29' east longitude and from 46° 20' north latitude to 46° 33'. The area of river basin witch is situated in the upstream of Nahand dam and hydrometric station is 219 kilometers.2.2. Phase space reconstructionOne way of characterizing dynamical systems is by the concept of phase space. The Takens theorem states that the underlying dynamics can be fully recovered by building an m-dimensional space wherein the components of each state vector Yt are correlated to observed values, which are discrete scalar time series, Xt={x1, x2,. ..xN} with N-observed values, with delay coordinates in the -dimensional phase space: (1)Where is referred to as the delay time and, for a digitized time series, it is a multiple of the sampling interval used, and m is termed the embedding dimension. The reconstruction of phase-space by plotting Xt against can show the presence of an attractor as a visual evidence for deterministic chaos in a given time series.Mathematical approaches for the reconstruction of the phase space diagram of chaotic behaviours may be carried out by one of the following

(i) Autocorrelation Function, ACF, (ii) Average Mutual Information, AMI.Correlation dimension is a nonlinear measure of the correlation between pairs lying on the attractor. The behavior of (correlation exponent or the slope of versus) provides one technique for determining the presence of chaos in a time series, such that (i) for stochastic processes, varies linearly with increasing m, without reaching a saturation value; (ii) for deterministic processes, the value of saturates after a certain value of m.3. Results and discussion3.1. Identification of low-dimensional chaos in the time seriesIn this study, the 39 years dataset of Nahandchai station is used. A visual assessment for the existence of chaotic behavior in the river flow time series may be obtained by the reconstruction of phase space diagram and a selection of results for those at Nahandchai station show chaotic behavior. This is suggestive of a possible existence of low-dimensional chaos in both of the dataset, in which the narrow dark band signifies strong determinism and the scattered band signifies the presence of noise in the data.Two methods are used to identify a possible existence of chaos in the river flow time series at Nahandchai station. Using the AFC method, the delay time τ is estimated for the time series at both of the stations at each timescale of the time series as the intercept with the x-axis of the curves by plotting the values of the ACF evaluated by the TISEAN package against delay times progressively increased from 1 to 100. The values of delay times are obtained as the zero intercepts of ACF equal to 55.The correlation function method is implemented by setting the embedding dimension values, m, from 1 to 20 and varying systematically the values of r from a low value to say 100. The result is shown in Fig. 1 for Nahandchai station. The method identifies the existence of chaos in the following ways: (i) By plotting logC(r)/log(r) versus log(r), the function values tend to fluctuate at low values of radius r, signifying their stochastic strength, but for higher values of r the function tends to find a plateaux, where the values of logC(r)/log(r) becomes saturated for each timescale value providing visual evidence for a deterministic behavior. (ii) The behavior of correlation function C(r) against radius r for values of increasing m for the time series, providing further evidence for deterministic chaos if the correlation function converges towards a single point underpinning the role of deterministic processes. (iii) The values D2(m) increase with increasing the embedding dimension values, m, up to a certain value but the existence of chaotic behavior is only underpinned if the values D2(m) saturate by reaching a plateau.3.2. Results by local prediction Local prediction algorithm is used to predict river flow time series at Nahandchai station. The procedure involves varying the value of the embedding dimension in a range, say 2–10, and estimating the value of correlation coefficient (R2) and Root Mean Square Error (RMSE). The embedding function with the highest coefficient of correlation is selected as the solution. 4. The slope of the line in figure (2-a) is the correlation dimension values and the embedding dimension values is shown in Figure (2-b). The saturation of the correlation dimension beyond a certain embedding dimension value is an indication of the existence of deterministic dynamics. The saturated correlation dimension is about 3.02 (D2=3.02). Results show that the best prediction is achieved when the embedding dimension is delay time=55 day and mopt=4.Fig. 2. Results of AFC: (a) Log C(r) Versus Log(r) for m= 1-20, (b) Relationship between embedding dimension and correlation dimension4. ConclusionsChaos theory with quantum theory and relativity is one of the most important discoveries of the last century. With review of time series caused by dynamical systems, such as behavior of rivers, behavior of system can be predicted by chaos theory. Dimensionality of a time series represents the level of complexity of the underlying system dynamics (and number of dominant governing variables), and therefore the above mentioned nonlinear dynamic and dimensionality-based classification certainly helps in identifying the appropriate structure and complexity of models. Sivakumar and Singh [6] have classified stream flow in the western United States according to correlation dimension to 4 classes.The nearest integer above the saturation value, fractals dimension, is generally considered to provide the minimum number of phase-space or variables necessary to model the dynamics of the attractor. The value of the embedding dimension at which the saturation of the correlation exponent occurs generally provides an upper bound on the number of variables sufficient to model the dynamics. Results of this study (D2=3.02) indicates low chaotic and less complex system. Number of variables necessary for model is 3 (fractals dimension=3). The optimum embedding dimension (mopt=4) from the nonlinear prediction method providing the presence of low-dimensional chaotic behavior in the river flow dynamics. The near-accurate predictions achieved for the runoff series time (correlation coefficient of about 0.87 and root mean square error of about 0.08) indicate the appropriateness of the chaotic dynamical approach for characterizing and predicting the runoff dynamics at the Nahandchai catchment.5. References[1] Domenico, M., Ghorbani, M. A., "Chaos and Scaling in Daily River flow", arxiv, 2010.[2] Ghorbani, M. A., Kisi, O., Alinezhad, M., "A Probe into the Chaotic Nature of Daily Streamflow Time series by Correlation Dimension and largest Lyapunov Methods", Applied Mathematical Modeling, 2010, 34, 4050-4057.[3] Islam, M. N., Sivakumar, B., "Characterization and Prediction of Runoff Dynamics: A Nonlinear Dynamics View", Advances in Water Resources, 2002, 25, 179-190.[4] Sivakumar, B., Berndtsson, R., Person, M., "Monthly Runoff prediction Using Phase Space Reconstruction", Hydrological Sciences-Journal-des Sciences Hydrologiques, 2001, 46 (3), 377-387.[5] Regonda, S. K., Sivakumar, B., Jain, A., "Temporal Scaling in River Flow: Can it be Chaotic?", Hydrological Sciences-Journal-des Sciences Hydrologiques, 2004, 49 (3), 373-385.[6] Sivakumar, B., Singh. V. P., "Hydrologic system complexity and nonlinear dynamic concepts for a catchments classification framework", Hydrology and Earth System Sciences, 2012, 16, 4119-4131.

Chaos theory studies complex and nonlinear dynamic systems and phenomena that their behavior seems to be stochastic and irregular at first glance, but in fact these systems are ruled by certain laws and with a deeper look, a kind of periodicity and regularity are evident in them. Sensitivity to initial conditions starting behavior, instability, non-linearity and non-periodic behavior, define a chaotic system. In addition, in chaotic hydrological systems, the short term time series can be extracted through analysis of long term time series. Furthermore, Information and relationships of the system can be achieved without any need to find the rules and governing dynamic relationships. Since the lakes water level fluctuations are dynamic and management of these ecosystems requires exact data at different intervals, chaos theory can play a unique role in acquiring information from this phenomenon. In recent years, Urmia Lake water level as an international wetland has been significantly reduced. Considering to the importance of this lake in Iran and the world, the purpose of this paper is to study the daily fluctuations in the Urmia Lake water level during the past 44 years using the concepts of chaos theory. To study the chaotic behavior of this phenomenon, correlation function, which is one of the exact methods to study these behaviors, was used; therefore, after computing the delay time and reconstructing the phase space, embedding dimension is determined using correlation function and false nearest neighbor algorithm, eventually correlation graph's slope is calculated. The non-integer digit result for this slope is an important index to identify this system as a chaotic system. In addition, studying the other indexes of chaotic behavior of water level such as positive Lyapunov exponent and broad band Fourier power spectrum show the quite chaotic behavior of time series studied.

Accurate estimation of the sediment outflow is one of the most important factors in water development projects. Estimation of sediment yield is required in a wide spectrum of problems such as: design of reservoirs and dams, transport and deposition of sediment in channel networks and water pollution. It is difficult to appoint governing equations of suspended sediment because of different parameters effects, and comparative mathematical models usually dont have enough accuracy. In this study, chaos theory was applied to model suspended sediment of Lighvan river watershed. The application of chaos theory in hydrology has been gaining considerable interest in recent times. An attempt is made in this study to characterize and predict daily suspended sediment load series using ideas gained from chaos theory during 30 years from the Lighvan River. To reconstruct phase space, the delay time and embedding dimension are needed and for this purpose,autocorrelation function and algorithm of false nearest neighbors are used. Correlation dimension method is applied for probe chaotic behavior of daily sediment load and obtained low correlation dimension chaos express chaotic behavior in the time series under investigation. Local prediction algorithm is used for prediction time series that results illustrate good and acceptable accuracy of chaos theory to predict daily suspended sediment load.