Homogeneous cosmological models, self-similar motion of self-gravitating
gas and motion of gas with homogeneous deformation have important
applica- tions in the theory of evolution of the universe. In particular
they can be applied to the theory of explosions of stars, formation of
galaxies, pulsation of alternating stars etc. The equations of general
relativity and Newtonian gas dynamics in the cases mentioned above are
reduced to systems of a finite (but quite large) number of ordinary
differential equations. In the last two decades these multi-dimensional
dynamical systems were and still are being analyzed by means of
traditional analytic and numerical methods. Important dynamical modes of
some solutions were thus established. These include oscillatory modes of
the space-time metric near a cosmological singularity, self-similar
motion of self-gravitating gas with a shock wave and an expanding cavity
inside (as in an explosion of a star), collapse of an ellipsoid of
self-gravitating dust into a disc and others. However the multi-
dimensional dynamical systems in question are so complex, that a
complete analysis of all dynamical modes of the solutions by means of
well-known tra- ditional analytic methods does not seem feasible.
Therefore the development of effective methods of qualitative analysis
of multi-dimensional dynamical systems and their application to the
problems of astrophysics and gas dynamics previ- ously unsolved by
traditional methods becomes especially urgent.