Antonio Pumari\~no, Claudia Valls
Instability in two Hamiltonian families
(508K, LATeX 2e)

ABSTRACT.  >From the time of the pioneer Poincar\'e's essay up to the present
days, chaos in conservative dynamics has been identified with the
presence of heteroclinic motions in transversal cross sections to the 
flow. The existence
of this unlimited dynamical richness leads, in an unmistakable way, to
the instability of the studied system. V. I. Arnold discovered that,
surprisingly, these situations often arise in a persistent way when
an integrable Hamiltonian system is perturbed.
The global strategy designed by Arnold was based on the control of the
so-called splitting of separatrices, which takes place when a parametric family
of perturbations of the initial integrable system is considered. The
method used by Arnold furnished orbits drifting along invariant objects 
and therefore giving rise to the presence of (nowadays called) Arnold diffusion.
>From the quantitative point of view, those events were observed for an open,
but small, set of parameter values.
Besides proving the existence of Arnold diffusion for a new family of three
degrees of freedom Hamiltonian systems, another goal of this paper is not
only to show how Arnold-like results can be extended to substantially larger
sets of parameters, but also how to obtain effective estimates on the 
splitting of separatrices size when the {\em frequency} of the perturbation
belongs to open real sets.