Example 6 - A Chaotic Example
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Data File: Full Logistic Map.csv, Model Files: Full Logistic Map.md and Full Logistic Map2.md
Settings file: Full Logistic Map.work and Full Logistic Map2.work (automatically loads into ChaosHunter when you load the model file)
The Full Logistic Map is a well known equation that produces a chaotic time series:
X t+1 = 4(X t – X t2)
where X t means the value of X at time t.
Certain values of X 0 will cast the series into chaos, e.g. 0.876. We wanted to see if the ChaosHunter could find the Full Logistic Map equation, given the time series.
Since there is only one time series, we could not select both an input and an output, so we made the time series the output, and set the optimization set to start at the second value in the series (the second row of data). For the input, we selected the 1 period lag of the time series. For good measure, we also checked the box allowing the Chaos Variable to become an input, and we set the initial value to 0.876. The ChaosHunter can choose either one of these as an input variable, since they are the same.
We made two different runs which resulted in two different equations. We leave it to the reader to apply high school algebra to verify that both equations simplify to the Full Logistic Map Equation.