Added new tools to first re-interpret and confirm findings from the research paper (ref. 1 and 2) and second to have a way to compute Lyapunov exponents from any (including lottery) data/time series.

In 1984 a paper titled "DETERMINING LYAPUNOV EXPONENTS FROM A TIME SERIES" by Alan WOLF~-, Jack B. SWIFT, Harry L. SWINNEY and John A. VASTANO
outlined a new algorithm for Lyapunov Exponent Spectrum calculations.

Excerpt from the abstract:

"
We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic."

Subsequently, Fortran code was ported to C++ and incooperated into Chaos Toolkit. More tests are needed to verify correctness of its implementation.

And although implementation details are complex, Lyapunov exponents are well known and serve in analyzing non-linear data series random/chaotic properties.

Using the new toolkit and RPS game model (ref. 2) following two graphs below were generated in order to demonstrate that the RPS Game exhibits chaos (Average Lyapunov Exponents > 0)

Graph 1. Probability Values for Player One RPS Game

Graph 2. Lyapunov Exponents in RPS Game

References:

1. DETERMINING LYAPUNOV EXPONENTS FROM A TIME SERIES Alan WOLF~-, Jack B. SWIFT, Harry L. SWINNEY and John A. VASTANO Department of Physics, University of Texas, Austin, Texas 78712, USA

2. Chaotic time series prediction for the game, Rock-Paper-Scissors by Franco Salvetti, Paolo Patelli and Simone Nicolo

http://www.francosalvetti.com/FrancoSalvetti_in_press.pdf

Complete source code files:

Last Edited: June 29, 2009, 12:25 am