An older Javascript animation
(where your browser actually integrates the differential equations). Here, Mathematica was solving
the system numerically with NDSolve.
The big open question here is "Is this system chaotic on a set of positive measure".
The common believe is yet, but we are mathematically just in an infantile stage and have
currently no idea how to prove this mathematically. The flow is a differential equation
on a three dimensional energy surface in a four dimensional manifold (the product of two
two dimensional cylinders). While chaos can not happen in two dimensions by Poincaré-Bendixon,
this is possible in three dimensions. If dA(t) is the 4x4 Jacobean matrix of the time t map,
the conjecture is that some matrix entries grow exponentially fast on a set of initial conditions
which have positive volume. This is a problem belongs to smooth ergodic theory
which is a field of mathematics which mixes differential equations, linear algebra,
calculus and probability theory. The problem is equivalent to the statement
"Does the system have positive metric entropy?". The entropy notion is a dynamical
spin-off of the Shannon entropy S we have seen in class. "Chaos" essentially means that
the entropy of finite partitions grows exponentially with time. It means that the system actually
behaves like a true random number generator in some parts of the phase space.
I worked on this in my thesis.
To the background music:
"Liberators" from album "Vengeance" (2015) by Epic Score.