Nonlinear Phenomena in Complex Systems, 1999, Vol.2, No.2, pp.63--71

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

1999, Volume 2, Number 2, pp.63--71

Non-Integrability and Infinite Branching of Solutions of 2DOF Hamiltonian Systems in Complex Plane of Time.
Vassilios M. Rothos and Tassos C. Bountis

It has been proved by S.L. Ziglin, 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. 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.

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