The methods which one uses to understand or even construct <PERSON>'s theorem do indeed belong to the wide class of methods which are used also in chaos theory, but the connection is rather loose. I will describe how you can understand even this loose connection.
When we have a dynamical system in classical mechanics, we need to analyze whether it is integrable or not. Integrable means that along the trajectories of the system, we have enough integrals of motion such as angular momentum or energy so that we can express all the future positions and velocities in terms of the initial data in terms of quadratures (possibly implicit formulas featuring definite integrals).
Consider the example of a particle moving in 1 dimension in a time-independent potential $V(x)$. For this particle, we know that it conserves energy along its motion $$E = \frac{1}{2}m \left( \frac{dx}{dt} \right)^2 + V(x)$$ We can thus express its velocity in terms of the initial energy and position $$\frac{d x}{ dt} = \sqrt{2 (E - V)/m}$$ As long as $dx/dt \neq 0$ you can formally rewrite this system as $$\frac{d x}{\sqrt{2 (E - V)/m}} = dt$$ And after integration you end up with $$\int_{x_0}^{x} \frac{d x}{\sqrt{2 (E - V)/m}} = t - t_0$$ this is what would be called a solution in terms of quadratures.
For a higher-dimensional classical-mechanical (Hamiltonian) systems, we have theorems such as Liouville-Arnold's theorem which tells us that if a suitable set of integrals of motion exists, then there exists a solutions in terms of quadratures. More specifically, for a system of $N$ degrees of freedom, one needs a set of $N$ integrals of motion which are independent and in involution via the <PERSON> bracket.
For bound motion, this ends up in the motion breaking up into $N$ independent periodic motions with $N$ generally independent "fundamental" frequencies. Consider a particle moving in a plane and a central potential:
The motion breaks up into 1) a periodic oscillation in the distance from the centre (the periastron and apoastron), and 2) the periodic rotation around the center. If these two frequencies do not match, we get a quasi-periodic motion, a trajectory which is not periodic in the sense of reaching the same point and velocity at a finite time but which still breaks up into a regular product of periodic motions.
However, if there is not a sufficient amount of integrals of motion, the motion gets chaotic. It is notoriously difficult to analytically prove that there is not a sufficient number of integrals of motion for the given set of trajectories in a dynamical system and that these trajectories will thus necessarily get chaotic. 453
There are some methods such as the <PERSON> integral (sometimes called Poincaré-Melnikov-Arnold integral) which can show that small layers of chaos in the phase space exist. (<PERSON> used something of this sort to prove chaoticity of the three body problem.) But actual investigations in chaos are mostly built around numerical or semi-numerical approaches.
But now I am getting to <PERSON>'s theorem; you can actually sort of prove <PERSON>'s theorem by thinking about integrability.
The reason is that when you have a system which is "too integrable", the number of fundamental frequencies reduces. For instance, the motion of a particle in three dimensions will generically have three fundamental frequencies, but the motion in a spherically symmetric and stationary potential will have just two. This is due to the fact that stationarity will give you an energy integral of motion (due to <PERSON>'s theorem), and three angular momenta arise due to three rotation symmetries (think Euler angles, not polar coordinates $\vartheta, \varphi$). Not all the angular momenta are in involution but it does not stop them from acting as a constraint which restricts the freedom in which the fully integrable trajectory can wobble. Specifically, the conservation of the whole angular momentum vector restricts the motion to a plane as shown above. So, in spherical systems you end up with an "overly integrable" system which has effectively two degrees of freedom and only two fundamental frequencies.
Now comes the motion in the $1/r, r^2$ potentials. These have special additional symmetries which make the posses additional integrals of motion. 976