  In light filamentation induced by axicon-generated, powerful Bessel beams,
the spatial propagation dynamics in the nonlinear medium determines the
geometry of the filament channel and hence its potential applications. We show
that the observed steady and unsteady Bessel beam propagation regimes can be
understood in a unified way from the existence of an attractor and its
stability properties. The attractor is identified as the nonlinear unbalanced
Bessel beam (NL-UBB) whose inward H\"ankel beam amplitude equals the amplitude
of the linear Bessel beam that the axicon would generate in linear propagation.
A simple analytical formula that determines de NL-UBB attractor is given.
Steady or unsteady propagation depends on whether the attracting NL-UBB has a
small, exponentially growing, unstable mode. In case of unsteady propagation,
periodic, quasi-periodic or chaotic dynamics after the axicon reproduces
similar dynamics after the development of the small unstable mode into the
large perturbation regime.
