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Study of Statistical Behavior of Chaotic Maps and Design of Stochastic Models for Reconstruction and Prediction of Behavioral Patterns of Chaotic Systems

Jokar, Meysam | 2017

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 50156 (08)
  4. University: Sharif University of Technology
  5. Department: Mechanical Engineering
  6. Advisor(s): Salarieh, Hassan; Alasty, Aria
  7. Abstract:
  8. Chaotic time series analysis, study of statistical behavior of chaotic maps and eventually an attempt to reconstruction and prediction of dynamical and statistical properties of output data of chaotic systems using stochastic models such as Markov models and autoregressive-moving average models are the main purposes of the present research. Examples of chaotic time series abound in the output of economics, engineering systems, the natural sciences (especially geophysics and meteorology) and social sciences. An intrinsic feature of an output time series of a dynamic system is that, adjacent observations are dependent. Time series analysis is concerned with techniques for the analysis of this dependence and the development of dynamical and stochastic models is usually done for its data reconstruction. Sensitive dependence of the solution on initial conditions is one of the most important features of chaotic systems, but their statistical behavior is independent on initial conditions. Also chaotic systems -as an important branch of nonlinear dynamical systems- are very similar to stochastic systems, especially in the high dimensions. Perhaps the most important similarity between these two types of systems is the existence of invariant probability density functions as a criteria for statistical behavioral. According to the above points, the stochastic modelling of chaotic output data of chaotic systems and in particular, the Markov model is very useful and important. Sometimes, an equation can be very useful for prediction and control, especially when the main system model is not available. In this thesis, after introducing chaotic time series analysis methods such as fractal dimensions and Lyapunov exponent, two methods are presented for stochastic modelling of these series. The first method is the design of different orders of Markov models with state transition matrix extraction approach and the second method is the design of ARMA models. It will be seen that the Markov models are successful in statistical reconstruction and prediction, but the ARMA models are only useful for reconstruction. However, their advantage is that they obtain a linear stochastic difference equation
  9. Keywords:
  10. Time Series ; Prediction ; Markov Model ; Chaos Theory ; Stochastic Modeling ; Statistical Reconstruction ; Autoregressive-Moving Average Models

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