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Abstract:
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Modeling of chaotic systems, based on the output time series, is quite
promising since the output often represents the characteristic behaviour of
the total system. It has been an interesting topic for researchers over the past
few years. So far, some methods are developed for the identification of
chaotic systems. Because of the intense complexity of chaotic systems, the
performance of existing algorithms is not always satisfactory. Application of
chaotic system theory to socially relevant problems like environmental
studies is the need of the hour Neural networks have the required self-learning capability to tune the
network parameters (i.e. weights) for identifying highly non-linear and
chaotic systems. In the present work, effectiveness of modeling a chaotic
system using dynamic neural networks has been demonstrated. From the rich
literature available for non-linear modeling with neural networks, the
Recurrent Neural Network (RNN) structure is selected. The Extended
Kalman Filter (EKF) algorithm is used to train the RNN. Further, the
Expectation Maximization algorithm is used to effectively arrive at the initial
states and the state covariance. Particle filter algorithm with its two important
variants namely Sampling Importance Resampling (SIR) and Rao
Blackwellised algorithms are also used for training the given RNN. Four
standard chaotic systems, Lorenz, Rossler, Chua and Chen, are modelled
with the three algorithms. The best algorithm is found to be EKF-EM based
on the least mean square error criterion. Validation of RNN model with EKFEM
algorithm is done in time domain by Estimation of embedding
dimension, Phase plots, Lyapunov Exponents, Kaplan -Yorke dimension and
Bifurcation diagrams. Analysis of the chaotic systems is also performed in
the transform domain using Fourier, Wavelet and Mapped Real Transforms.
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Natural chaotic systems are analyzed based on the selected model structure
and training algorithm, taken for analysis. Sunspot, Venice Lagoon and
North Atlantic oscillations are the three of the natural chaotic systems
modelled with the selected RNN model structure and EKF-EM algorithm. |