Homogeneous models in general relativity and gas dynamics

The paper begins with a short survey of results on non-trivial models (that is, those that are not integrable analytically) in general relativity and gas dynamics. The investigation of these models is carried out by the methods of the qualitative theory of many-dimensional dynamical systems, using geometrical and topological ideas. The first section deals with the results of research on the evolution of homogeneous cosmological models with a hydrodynamic energy tensor – the impulse about a singularity. In the second section similar models are applied to the study of the complex oscillating regimes of a classical ideal compressible fluid. The Appendix contains new, unpublished results due to one of the authors, describing stochastic perturbation of a completely integrable Toda chain.