Methods in the Qualitative Theory of Dynamical Systems in Astrophysics and Gas Dynamics

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.

I. Methods of Qualitative Analysis of Multi-Dimensional Dynamical Systems.- 1. Prerequisites from the Qualitative Theory of Two-Dimensional Dynamical Systems.- 2. Analysis of Degenerate Critical Points of a Dynamical System.- 3. Maximally Non-Degenerate Compactification of a Dynamical System.- 4. Separatrix Approximation Method for the Trajectories of a Dynamical System.- II. Qualitative Theory of Homogeneous Cosmological Models Without the Motion of Matter.- 1. Equations of the General Theory of Relativity.- 2. Classical Solutions to the Equations of the General Theory of Relativity.- 3. General Properties of Homogeneous Cosmological Models.- 4. Transformation of the Hamiltonian System.- 5. Cosmological Models of Types I and II.- 6. Cosmological Models of Type IX.- 7. Analysis of Cosmological Models of Types VIII, VII0 and V10.- 8. Transformation of the Dynamical System for Homogeneous Cosmological Models of Class B.- 9. Several General Properties of the Dynamics of Homogeneous Cosmological Models of Types III, IV, VI and VII.- 10. Analysis of Several Special Properties of Homogeneous Cosmological Models of Types V, VII, III, VI and IV.- III. Qualitative Theory of Homogeneous Cosmological Models with the Motion of Matter and Electromagnetic Fields.- 1. Einstein’s System of Equations for the Homogeneous Cosmological Model of Type IX with the Motion of Matter.- 2. Transformation of the Dynamical System.- 3. Power Asymptotics. Typical States of the Metric in the Early Stages of the Expansion of Space.- 4. Combinatorial Model of the Oscillatory Mode.- 5. Several Common Properties of the Dynamics of Homogeneous Cosmological Models with the Motion of Matter.- 6. Homogeneous Cosmological Model of Type IX with Electromagnetic Fields.- IV. Self-Similar Spherically Symmetric Solutions for the General Theory of Relativity.- 1. Einstein’s System of Equations for Spherically Symmetric Self-Similar Solutions.- 2. Analysis of the Dynamical System.- 3. Transformation of Self-Similar Solutions in Various Coordinates.- 4. The Problem of Breakdown of Equilibrium of a Star in the General Theory of Relativity.- 5. Self-Similar Solutions with Expanding and Converging Shock Waves.- V. Self-Similar Motion of Self-Gravitating Gas in Stars.- 1. Resolution of Singularities of the Dynamical System.- 2. Asymptotics of Gas Moving Away from the Center.- 3. Analysis of the Dynamical System on the Components of the Boundary ?2 and ?8.- 4. Self-Similar Accretion of Self-Gravitating Gas to the Center.- 5. New Solutions in the Model of Stellar Explosions.- 6. Analysis of Models of Explosions in Stellar Envelopes.- 7. Self-Similar Solutions with Converging Shock Waves.- VI. Self-Similar Rotation of an Ideal Gas.- 1. Definition of Self-Similar Rotation of an Ideal Gas.- 2. Algebraic Integrals for the Self-Similar Rotation of an Ideal Gas.- 3. Exact Self-Similar Power Solutions.- 4. Analysis of the Dynamical System.- 5. Self-Similar Expansion of Rotating Gas.- 6. Several Self-Similar Solutions with ? = 2.- VII. The Dynamics of a Gaseous Ellipsoid.- 1. Equations of Motion of a Non-Gravitating Gaseous Ellipsoid.- 2. Oscillatory Mode of Expansion of a Rotating Gas Cloud into Vacuum.- 3. Analysis of a Problem in the Theory of Shallow Water.- 4. Equations of Motion of a Gravitating Gaseous Ellipsoid.- 5. Transformation of the Hamiltonian System.- 6. Oscillatory Mode of Motion with Negative Energy.- 7. On the Impossibility of Collapse of a Gravitating Gaseous Ellipsoid in the Presence of Rotation of the Gas.- 8. Oscillatory Mode of Motion with Positive Energy.- 9. Concluding Remarks.- VIII. The Dynamics of Perturbations of the Periodic Toda Lattice.- 1. Hamiltonian Perturbations of the Toda Lattice.- 2. Separatrix Approximation of the Oscillatory Mode.- 3. Hamiltonian Systems Connected with Simple Lie Algebras.- 4. Non-Linear Oscillatory Modes in Systems of Hydrodynamical Type.