Vassilios M. Rothos, Tassos C. Bountis
Non-Integrability and Infinite Branching of Solutions of 2DOF
Hamiltonian Systems in Complex Plane of Time
(37K, LaTex)

ABSTRACT.  It has been proved by S.L.Ziglin (1982), for a large class 
of 2-degree-of-freedom (d.o.f) Hamiltonian systems, 
that transverse intersections of the invariant
manifolds of saddle fixed points imply infinite branching of 
solutions in the complex time plane and the non--existence of a second
analytic integral of the motion.  Here, we review in detail our recent results,
following a similar approach to show the existence 
of infinitely--sheeted solutions for 2 d.o.f.
Hamiltonians which exhibit, upon perturbation, subharmonic bifurcations 
of resonant tori around an elliptic fixed point (Bountis and Rothos, 1996). 
Moreover, as shown recently, these Hamiltonian systems are non--integrable 
if their resonant tori form a dense set. 
These results can be extended to the case where 
the periodic perturbation is not Hamiltonian.