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Lyapunov Exponents Spectrum Algorithms, Implementation and Verification

Published:

Last Edited: July 5, 2009, 11:25 am

Happy to report that tools development to compute Lyapunov exponents spectrum data is now complete:

3 different methods now yield the same numerical results and are in agreement with other sources

Henon map (discrete dynamical system) was used to test validity of software algorithm models.

Also RPS (Rock Paper Scissors game) computations were re-done yielding correct Lyapunov Exponent spectrum for the game, verifying that the game can be modeled by chaotic attractor (just the same as Henon map) despite its random probabilistic trajectories phase space.

In the summary following 3 different algorithms were used:

1. Wolf method for system of differential equations (see ref. 1 appendix A and ref. 2)

Lyapunov Exponents Using Wolf ODE method

2. Wolf method for discrete time series (without ODE model)  (see ref. 1 appendix B)

Lyapunov Exponents using Wolf method (without ODE model)

3. Sano Sawada method (see ref 3)

Lyapunov Exponents using Sano Sawada method

In all above cases Lyapunov exponents converge and oscillate around .42, in the full agreement with quoted sources.

In addition a new source of algorithms and application tools can be found in ref. 4

Hope is that we can use these tools to run thru some discretized form of lottery data , let it be frequency/number transition or some other reformulated time series.

As the tool-set is expanding we will be adding computation algorithms and models for Local Lyapunov Exponents, Entropy and Entropy Filtering.

Attempt is to first verify that we follow to best degree possible existing body of research and agree on the numerical results before tackling on lottery number distributions. 

References:

1. Determining Lyapunov exponents from a time series by Wolf, Alan; Swift, Jack B.; Swinney, Harry L.; Vastano, John A. [Physica D: Nonlinear Phenomena, Volume 16, Issue 3, p. 285-317.] 

http://adsabs.harvard.edu/abs/1985PhyD...16..285W

2. Numerical Calculation of Largest Lyapunov Exponent by J. C. Sprott

http://sprott.physics.wisc.edu/chaos/lyapexp.htm

3. Measurement of the Lyapunov Spectrum from a Chaotic Time Series by M. Sano and Y. Sawada Research Institute of Electrical Communication, Tohoku University, Sendai 980, Japan

http://prola.aps.org/abstract/PRL/v55/i10/p1082_1

4. TISEAN - Nonlinear Time Series Analysis by Rainer Hegger, Holger Kantz and Thomas Schreiber 

("TISEAN is a software project for the analysis of time series with methods based on the theory of nonlinear deterministic dynamical systems, or chaos theory, if you prefer. It has grown out of the work of our groups during the last few years.")

http://www.mpipks-dresden.mpg.de/~tisean

Entry #22

Comments

1.
johnph77Comment by johnph77 - July 5, 2009, 12:47 pm
No idea whether this will help or not and I don't know whether you're even aware of its existence, but.....

http://www.johnph77.com/math/lf.html

gl

j

2.
edgeComment by edge - July 5, 2009, 1:49 pm
your reference is fantastic thank you!
3.
Comment by pumpi76 - July 5, 2009, 4:00 pm
man, this sounds awesome...keep it up, do not waiver...and don't ever abandon this....
4.
Comment by joker17 - July 5, 2009, 6:26 pm
It's also important to note that the indicator which solely relies on the parameters within the 0.6 illumination, might very well accumulate thus exponentially re-aligining the factors of 17.3 filtration sequences and mathematical positioning.

In other words, probability and statistical data was the foundation from where the actual primers were established when only 5% of the total oscillating wave frequency charts were examined. Furthermore, it's relation with the deflection analysis showed erratic yet impressive findings. The algorithms and peak charts also relayed mounting evidence which showed the specifications were sound, meaning much of the empirical evidence was undermined early on.

Hopefully, further research will be analyzed with scrutiny so that future numerical properties won't enhance the models without producing erroneous readings. Full degree tables can be found on the net which yield more accurate results than just the 13.7 differentials, and clearly doesn't elude the variance charts. Factoring in the variables and constants, many of the linear chaos programs might help the user identify the computations and permutations.
5.
edgeComment by edge - July 5, 2009, 6:50 pm
funny stuff:) contemplated to lock comments section, but I am all about democracy and good ol' fun as well:)
6.
Comment by joker17 - July 5, 2009, 10:04 pm
C'mon...just messin dude. I just couldn't help myself...lol

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