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Cell["C & Q Chaos IV - Phenomenology of broken integrability", "Title",
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Cell[TextData[{
 "Last time we explored the concept of what is Liouville integrability, and \
we started discussing what is the route to it being broken. The key element \
were the homoclinic/heteroclinic orbits and the tangles that they created \
upon perturbation. This explained the weird phenomenon that we saw around the \
unstable equilibrium in the phase space of the periodically driven \
mathematical pendulum. \n",
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Consider the unperturbed mathematical pendulum. Near x=0, it is almost a \
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This system has an angular frequency of \[OpenCurlyDoubleQuote]1\
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What are these new structures? Why do they occur in different parts of phase \
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 Let us take a look.\
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Cell["\<\
Interesting stuff! The perturbation has created a nonlinear pendulum within \
the nonlinear pendulum! The motion librates around a periodic orbit (there \
are two copies of the orbit for x>0 and x<0). There are also unstable \
periodic orbits along x=0, and roughly p = +-1.2. 

Obviously, the unstable periodic orbits also allow for a homoclinic tangle! \
Let\[CloseCurlyQuote]s try to take a look.\
\>", "Text",
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The tangle and the chaotic layer can be a very subtle feature but it is \
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Now consider this, however. We try to drive the oscillator with a frequency \
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frequencies is ",
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phases ",
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At a generic point, it has an infinite number of harmonics with respect to \
\[Psi]. However, the magnitude of the coefficients asymptotically drops \
exponentially since the function is smooth, so only a couple of the first \
ones matter. (Do note that the sequence may not be strictly monotonous.) \
Multiplied by the Sin[\[Tau]] term, there is a \
\[OpenCurlyDoubleQuote]persistent\[CloseCurlyDoubleQuote] term in the \
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The chains on the Poincar\[EAcute] surfaces of sections correspond to \
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dynamics is dominated by the perturbation. The  ~ Cos[2\[Psi] +\[Tau]] term \
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This is why we see a similar structure in the two problems. \
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(2006). ",
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