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The standard map is indeed a standard map, the canonical demonstration of how a conservative dynamical system breaks down into chaos as it becomes nonlinear. While strange attractors might have the best press, conservative systems have wonderfully complex dynamics. So what are the lyapunov exponents of the standard map Even if you, dear reader, do not understand anything of the previous I hope the following colorful fractal images might be pleasant viewing.
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.
What about the color inside the non-chaotic blue areas The color differences are caused due to different orbits circling the central fixed point at different speeds - there is divergence of orbits here too, just not very fast. Properly, the lyapunov exponent should be 1 since the divergence is not exponential (I think). This suggests that the real exponent actually increases from 1 to a finite value at the border between the regions. But the border is the usual kind of Cantor set times reals you get from Smale horseshoes, making the exponent as function of the position a very wild function - discontinous, oscillating like ceil(sin(1/x)) as one approaches a boundary - and the boundaries form a fractal set. To make matters worse, the KAM theorem shows that the chaotic and non-chaotic regions for some values of K both have non-zero measure, producing areas of positive measure where the lyapunov exponent function is nonzero in a dense set. So it is not just discontinous here and there, but across entire areas.
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