  The so-called Dixon system is often cited as an example of a two-dimensional
(continuous) dynamical system that exhibits chaotic behaviour, if its two
parameters take their value in a certain domain. We provide first a rigorous
proof that there is no chaos in Dixon's system. Then we perform a complete
bifurcation analysis of the system showing that the parameter space can be
decomposed into sixteen different regions in each of which the system exhibits
qualitatively the same behaviour. In particular, we prove that in some regions
two elliptic sectors with infinitely many homoclinic orbits exist.
