Could Different Attractors Explain Variations in Lyapunov Exponents?

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SUMMARY

The discussion centers on the hypothesis that variations in Lyapunov exponents of a 1-Dimensional time series may be explained by different types or numbers of attractors within the system. The user has employed methods such as Poincaré plots and Gaussian Mixture Models with the Calinski-Harabasz algorithm to identify clusters in the data. A suggestion was made to utilize Ruelle-Takens embedding as a potential method for attractor reconstruction. The user seeks confirmation on whether their observations can indeed be classified as attractors.

PREREQUISITES
  • Understanding of Lyapunov exponents in chaotic systems
  • Familiarity with Poincaré plots for analyzing dynamical systems
  • Knowledge of Gaussian Mixture Models and the Calinski-Harabasz algorithm
  • Basic concepts of attractor theory in chaos theory
NEXT STEPS
  • Research Ruelle-Takens embedding for attractor reconstruction
  • Explore advanced techniques for Lyapunov exponent calculation
  • Investigate other clustering algorithms for time series analysis
  • Learn about attractor types and their implications in chaotic systems
USEFUL FOR

Researchers and students in chaos theory, data scientists analyzing time series data, and anyone interested in the dynamics of chaotic systems and attractor analysis.

Nono92
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Hi,

I am a beginner and I don't speak very well... So I'm really sorry for my poor scientific language...

I work on 1-Dimension time series of a same system measured at different periods. In these periods, time series have different chaotic characteristics as their lyapunov exponent are different.
To explain these lyapunov exponent differences, I hypothesize the system could have different number and/or type of attractor(s). Do you think that this hypothesis could be right ?

In order to verify this hypothesis, I've tried to observe these attractors.
I don't know if it exists methods to observe it. Do you know methods to observe it ?

As I am a beginner I have tried several methods may be not adaptated.
These are the methods that I have applied :
So, I have plotted poincare plot, I have observed that at several moment there are rotations around a center.
I have calculated for each three followed points a gravity center, I have evaluated hom many cluster could be identified with a gaussian model mixture method with Calinsky Harabasz algorithm. And then I have applied gaussian model mixture with the number of clusters calculated previously on these gravity centers.

But I don't know if it is really attractors that I observe with these methods.
Could you help me ?
Thanks a lot

nono92
 
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