We have developed the {\it general method} for the description of {\it
separatrix chaos}, basing on the analysis of the separatrix map dynamics.
Matching it with the resonant Hamiltonian analysis, we show that, for a given
amplitude of perturbation, the maximum width of the chaotic layer in energy may
be much larger than it was assumed before. We apply the above theory to explain
the drastic facilitation of global chaos onset in time-periodically perturbed
Hamiltonian systems possessing two or more separatrices, previously discovered
(PRL 90, 174101 (2003)). The theory well agrees with simulations. We also
discuss generalizations and applications. Examples of applications of the
facilitation include: the increase of the DC conductivity in spatially periodic
structures, the reduction of activation barriers for noise-induced transitions
and the related acceleration of spatial diffusion, the facilitation of the
stochastic web formation in a wave-driven or kicked oscillator.Comment: 29 pages, 16 figures (figs. are of reduced quality, original files
are available on request from authors), paper has been significantly revised
and resubmitted to PR
