  In this work we introduce the generic conditions for the existence of a
non-equilibrium attractor that is an invariant manifold determined by the
long-wavelength modes of the physical system. We investigate the topological
properties of the global flow structure of the Gubser flow for the
Israel-Stewart theory and a kinetic model for the Boltzmann equation by
employing Morse-Smale theory. We present a complete classification of the
invariant submanifolds of the flow and determine all the possible flow lines
connecting any pair of UV/IR fixed points. The formal transseries solutions to
the Gubser dynamical system around the early-time (UV) and late-time (IR) fixed
points are constructed and analyzed. It is proven that these solutions are
purely perturbative (or power-law asymptotic) series with a finite radius of
convergence. Based on these analyses, we find that Gubser-like expanding
kinetic systems do not hydrodynamize owing to the failure of the
hydrodynamization process which heavily relies on the classification of
(non)hydrodynamic modes in the IR regime. This is in contrast to longitudinal
boost-invariant plasmas where the asymptotic dynamics is described by a few
terms of the hydrodynamic gradient expansion. We finally compare our results
for both Bjorken and Gubser conformal kinetic models.
