Producing the Pictures
To produce the "Lyapunov'' pictures, we label the axes of our graph by the variables a and b. For each point (a,b) in the rectangle, we define two "logistic'' maps:
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A(x) = ax(1-x).
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B(x) = bx(1-x).
Now, we compose the two maps in some sequence. This sequence is kept the same for the whole picture. For example, to produce the "swallows'' picture above, the sequence AB was used.
We choose a random value x in the unit interval to act as a starting point, then we iterate the composed map with that starting point. In the case of the "swallows'', we would iterate the composed map AB(x)=B(A(x)), yielding the sequence of values:
AB(x), AB(AB(x)), AB(AB(AB(x))), ...etc
The stability of the resulting system is measured by estimating its Lyapunov exponent. The value of the Lyapunov exponent is used to choose the colour for the pixel (a,b) in the picture:
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Positive values mean that the system is chaotic (in that tiny errors tend to be magnified). Points with these values form the backgrounds to the pictures above.
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Negative values indicate that the system is stable. Points with these values give rise to the coloured bands in the pictures.
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Values that approach negative infinity indicate that the system is super-stable. Points with these values give rise to the branching curves in the pictures.
Each of the pictures above corresponds to a different choice of the sequence of the maps A and B used in the iteration.
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