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In this paper we review a recently developed approximate method forinvestigation of dynamics of compressible ellipsoidal figures. Collapse andsubsequent behaviour are described by a system of ordinary differentialequations for time evolution of semi-axes of a uniformly rotating, three-axis,uniform-density ellipsoid. First, we apply this approach to investigate dynamicstability of non-spherical bodies. We solve the equations that describe, in asimplified way, the Newtonian dynamics of a self-gravitating non-rotatingspheroidal body. We find that, after loss of stability, a contraction to asingularity occurs only in a pure spherical collapse, and deviations fromspherical symmetry prevent the contraction to the singularity through astabilizing action of nonlinear non-spherical oscillations. The development ofinstability leads to the formation of a regularly or chaotically oscillatingbody, in which dynamical motion prevents the formation of the singularity. Wefind regions of chaotic and regular pulsations by constructing a Poincarediagram. A real collapse occurs after damping of the oscillations because ofenergy losses, shock wave formation or viscosity. We use our approach toinvestigate approximately the first stages of collapse during the large scalestructure formation. The theory of this process started from ideas of Ya. B.Zeldovich, concerning the formation of strongly non-spherical structures duringnonlinear stages of the development of gravitational instability, known as'Zeldovich's pancakes'. In this paper the collapse of non-collisional darkmatter and the formation of pancake structures are investigated approximately.We estimate an emission of very long gravitational waves during the collapse,and discuss the possibility of gravitational lensing and polarization of thecosmic microwave background by these waves.

Regular and chaotic dynamics of non-spherical bodies. Zeldovich's pancakes and emission of very long gravitational waves