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The lyapunov exponents tell how a small ball centered around a point is expanded and squeezed by the dynamics. Positive exponents imply directions where points are separated, negative exponents mean convergence. A conservative 2D map has just one exponent of interest since any expansion has to be counteracted by a equal contraction - the two exponents are equal but with oppisite signs.


As the map becomes more chaotic, thin bands near hyperbolic fixed points become chaotic as they become homoclinic points and surround themselves with tangles (I'm reminded of the neurofiliamentary tangles of Alzheimer's disease). These orbits show up as the red or yellow bands in the maps. The (0,0) point tangle is especially visible in the corners. To get real high-intensity chaos you need space, and hence the largest bands also have the biggest exponents. But as more and more periodic orbits break down we get space-filling chaos.


I approximated the exponent using a simple method: iterate a orbit xn starting from a point x0 together with another orbit starting nearby at x0+epsilon, and calculate the difference after a few iterations. To get better pictures I repeated this a number of times and with different number of iterations (in order to avoid artefacts due to the periodicity of some orbits).


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