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# Chaos and Fractal Toolkit expanded with Correlation Integral and Lyapunov analysis

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Last Edited: June 21, 2009, 7:58 pm

Re-encoded Mathematica notebook (ref. 1) into C++ classes, calculations were checked by hand and Henon attractor has emerged together with the exact Correlation Integral in the agreement with Fractal Dimension calculations from other sources. (approximately ~ 1.26) [Best Fit: Y = -1.36378 + 1.2656 X]

Correlation Integral is of the form Log[Corr] = D Log [R] and its importance lies in detecting chaos/fractal property in a data series.

Henon attractor has specific fractal dimension (approx 1.26) and exhibits chaotic orbits around an attractor (see below) these orbits are invariant in a sense that they exhibit certain pseudo periodicity (pseudo, as the points will not re-generate in the exact position in the number plane but rather do so on ever enlarging scale)

Next in this project is Lyapunov exponent, which measures how fast this "scale" expansion happens, if the rate is slow, the orbits can be of stable/periodic or chaotic type
(they can also start to "escape" to infinity or "converge" to point).

Hurst exponent will tell us how "smooth" trajectories are (and is related to fractal dimension), we already have its first manifestation in "Correlation Integral" in the Henon attractor, it i useful insofar as to be able to approximate one data series with another (ie Henon map with some n-dimensional embedding added)

Below are various images and output displays related to the above, much still has to be done, but serious progress has been made.

Best fit (linear) Best Fit: Y = -1.36378 + 1.2656 X
in the full agreement with independent calculation of Hurst exponent for Henon map attractor (~ 2.26)

Lyapunov exponents for various (random) sequences

1 periodic -0.103898
3 chaotic 0.565543
4 chaotic 0.259456
5 chaotic 0.8249
8 chaotic 0.465392
9 chaotic 0.381847
10 periodic -0.0727799
11 chaotic 0.669459
12 periodic -0.178313
13 periodic -0.0211324
14 chaotic 0.802993
15 chaotic 0.469818
16 chaotic 0.196232
17 chaotic 0.976126

Results (88 seconds)

Infinite: 0 (0.0%)
Point : 4 (22.2%)
Stable : 0 (0.0%)
Periodic: 4 (22.2%)
Chaotic : 10 (55.6%)

Attractor images (only stable (0 found in this particular trial), periodic and chaotic images are displayed (infinite and point have no visual representation possible)

References:

1.  Testing Chaos and Fractal Properties in Economic Time Series

http://www.internationalmathematicasymposium.org/IMS99/paper25/ims99paper25.pdf

2. Numerical "best fit" linear estimation to determine Correlation Integral were performed using GSL - GNU Scientific Library:

3. Images showcasing attractors and their Lyapunov exponents were generated using ideas and sample application found at:

http://www.technocosm.org/chaos/attr-part2.html

Entry #15