Application of artificial neural network approach to study the dynamic of chaotic systems

Abstract

Artificial Neural Networks(ANNs) have been widely used in many disciplines to perform various complex functions. They are used to represent non-linear maps that can predict the state of a system from a given set of measurement values. ANNs are also used to represent very complex systems that exhibit chaotic behaviour. A number of studies are focussing on the use of ANNs in capturing attractors of chaotic systems including their invariants like Lyapunov exponent and Correlation Dimension. However, these studies have not so far been able to completely represent the structural as well as parametric features of such systems. One of the reasons for this drawback is the presence of noise in the data generated by ANNs. Recent studies are also focussing on the use of ANN in noise reduction. The methods include techniques involving shadowing of true profiles.

In this thesis, we attempt to develop a new method which uses ANNs alongwith some noise reduction algorithms to overcome difficulties in capturing the qualitative features of chaotic attractors and their invariants. The method combines ANNs with noise reduction algorithms at appropriate levels that capture the metric and dynamic invariants of a chaotic time series. While a feedforward neural network is capable of capturing the dynamical and metric invariants of chaotic time series within an error of about 30%. use of noise reduction techniques such as Hammel's method or the Extended Kalman Filter can significantly improve their performance.

We have used the Logistic map and the Henon map as platforms to demonstrate, the application of the technique. The Logistic map is an example for a one dimensional difference equation whereas the Henon map is an example for a two dimensional difference equation. These maps exhibit chaotic behaviour for certain parameter values. A four layer ANN has been used to train the Logistic map and thc Henon map. The attractors of the chaotic systems have been reproduced using the ANN within certain error 1evel. The reproduced values deviated from the original values by about 30%. The Correlation dimension and Lyapunov exponents of the trained values were calculated and compared with the original values. Here also difference of about 30% to 40% from the original values were observed. This might be due to the noise contained in the data generated. We have applied noise reduction algorithms viz. Hammel's technique and EKF to reduce the noise from the ANN generated data. In the case of the Logistic map, the attractors obtained after noise reduction compare well with the original attractors, whereas in the case of Henon map the attractor obtained after noise reduction is not satisfactory. However, the Correlation dimension and Lyapunov exponents of the filtered data of both the systems are very close to the original values. This indicates that the error contained in the ANN generated data has been reduced significantly. We also analysed our results using another noise reduction method viz. the Local Projective method to reduce the noise in the ANN generated data.

We have compared the results obtained from all these noise reduction methods. This comparison shows that Hammel's technique and the EKF technique are good filters in this context also and are superior to the Local Projective method in reducing the error level in the ANN generated data. Hence we conclude that the hybrid technique of combining ANN training with noise reduction algorithm, not only reproduces the chaotic attractors and its parameters correctly, it also reduces the training time of the ANN. This technique can be used in signal processing, image processing, telecommunications etc.

This thesis consists of six chapters. The first Chapter is an introductory Chapter. The second and third Chapters give brief introduction to ANN and Chaos respectively. Chapter four deals with the application of ANN to chaotic systems viz. Logistic map and Henon maps. Chapter five is an important Chapter in this thesis. It explains the various methods and techniques for noise reduction in chaotic systems including the Hammel's technique, the EKF and the Local Projective methods. Applications of these techniques to ANN trained chaotic systems are discussed in this Chapter. Chapter six brings out the comparison of the results of our investigations.

The results of this investigations are published/ under processing in the following journals.

1 ) "Application of chaotic noise reduction techniques to chaotic data trained by ANN", Accepted for publication in Sadhana, Indian Academy of Sciences, Bangalore, 2001. 2) "Application of noise reduction algorithms to a chaotic system trained by ANN". Resubmitted to IEEE Signal Processing Letters, USA.