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Title
Abstract
Key Takeaways
Figures
Introduction
Time Series Analysis
Conclusion
References

Some Aspects of Chaotic Time Series Analysis

Profile image of V. RubanovV. RubanovProfile image of Yury SavitskyYury Savitsky

2001

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Abstract

We address two aspects in chaotic time series analysis, namely the definition of embedding parameters and the largest Lyapunov exponent. It is necessary for performing state space reconstruction and identification of chaotic behavior. For the first aspect, we examine the mutual information for determination of time delay and false nearest neighbors method for choosing appropriate embedding dimension. For the second aspect we suggest neural network approach, which is characterized by simplicity and accuracy.

Key takeaways
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  1. The largest Lyapunov exponent, a positive value, measures divergence between chaotic trajectories.
  2. Neural networks can simplify and enhance the accuracy of estimating the largest Lyapunov exponent.
  3. Mutual information and false nearest neighbors are used to determine optimal time delay and embedding dimension.
  4. The proposed embedding dimension for Henon data is 3, and for Lorenz data is 5.
  5. This work outlines methods for state space reconstruction and chaotic behavior identification.
Figures (5)
Figure 1. Functional diagram of the data processing Dealing with one dimensional time series we must first of all perform state space reconstruction. It is based on an embedding theorem [2], which guarantees that a full knowledge of the behavior a system is In the Fig. 2 and 3 are represented the graphics of mutual information for Henon and Lorenz attractor respectively. As can be seen from the Fig.3 the first minimum of the mutual information is 0.16 for the Lorenz data. For the Henon data we can’t define a minimum. In this case we can take T =1 directly from the Henon equation.
Figure 1. Functional diagram of the data processing Dealing with one dimensional time series we must first of all perform state space reconstruction. It is based on an embedding theorem [2], which guarantees that a full knowledge of the behavior a system is In the Fig. 2 and 3 are represented the graphics of mutual information for Henon and Lorenz attractor respectively. As can be seen from the Fig.3 the first minimum of the mutual information is 0.16 for the Lorenz data. For the Henon data we can’t define a minimum. In this case we can take T =1 directly from the Henon equation.
Figure 7. I - the evolution of distance between two nearby orbits for Lorenz attractor; II - regression line
Figure 7. I - the evolution of distance between two nearby orbits for Lorenz attractor; II - regression line

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References (6)

  1. REFERENCES
  2. H. Kantz. A robust method to estimate the maximal Lyapunov exponent of a time series. Phys. Lett. A 185, 77 (1994).
  3. F. Takens. Detecting Strange attractors in fluid turbulence. Springer-Verlag, Berlin, 1981.
  4. A. Fraser, H. Swinneg. Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33, 1134 (1986).
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  6. Vladimir Golovko, Yury Savitsky, Nikolay Maniakov. Modeling Nonlinear Dynamic using Multilayer Neural Networks. Proceedings of the Workshop Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications. (IDAACS'2001), Foros, Ukraine, July 1-4 2001, PP. 197-202.

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