  This paper presents a more complete version than hitherto published of our
explanation of a transition from regular to irregular motions and more
generally of the nature of a certain kind of deterministic chaos. To this end
we introduced a simple model analogous to a three-body problem in the plane,
whose general solution is obtained via quadratures all performed in terms of
elementary functions. For some values of the coupling constants the system is
isochronous and explicit formulas for the period of the solutions can be given.
For other values, the motions are confined but feature aperiodic (in some sense
chaotic) motions.
  This rich phenomenology can be understood in remarkable, quantitative detail
in terms of travel on a certain (circular) path on the Riemann surfaces defined
by the solutions of a related model considered as functions of a complex time.
This model is meant to provide a paradigmatic first step towards a somewhat
novel understanding of a certain kind of chaotic phenomena.
