We analyze the properties of the solutions of a dynamical system when 
analytically continued to complex time. These solutions are generally 
multi-valued functions and the evolution in real time corresponds to a 
path on their  Riemann surface. The complexity of the dynamical system 
is seen to be intimately related to the complexity of the Riemann surface 
of the solutions. We illustrate  these ideas on a toy example provided by 
a three-body problem in the plane. 
Using these techniques we can we prove the existence of an open region in 
phase  space where the set of separatrices (collision manifolds) is dense. 
This gives  rise to a sensitive dependence of the solutions on the initial 
conditions,  although Lyapunov exponents are zero. The mechanism that 
gives rise to the increase of complexity will be thoroughly explained. 
When the coupling constants  of the model are rational numbers, we prove 
that all orbits are periodic and  stable, in fact the system is isochronous. 
The period of the orbit changes as a function of the initial data, and we are 
able to provide explicit formulas for  this dependence, which involve in 
some cases  the expansion in continued  fraction of the coupling constants. 
We observe that although the system under consideration is integrable, 
its dynamics can be very complicated, featuring  even sensitive dependence 
on the initial data. We believe  the mechanism  described in detail in this 
toy  model to be present in a general class of  dynamical systems described 
by  ordinary differential equations. 
The results  reported in this talk have been obtained in collaboration  with 
F. Calogero (Roma La Sapienza),  P. Santini (Roma La Sapienza) and
M. Sommacal (SISSA Trieste).