Osypova O. Integral manifolds and decomposition of systems of multiscale linear singularly perturbed equations

The thesis is devoted to the study of the method of decomposition of linear singularly perturbed systems based on the method of integral manifolds of fast and slow variables. The main idea of the splitting scheme is to select a group of slow variables of the studied system with its subsequent reduction to a special form in which the subsystem of slow variables does not contain fast variables. A method for constructing a sequence of nondegenerate substitutions of variables that lead the original singularly perturbed system to a set of independent subsystems is developed. Sufficient coefficient conditions for the initial system have been found, under which splitting transformations exist and are unique. An explicit form of the splitting transformation for systems of linear singularly perturbed differential equations with two and many small parameters is obtained. Since the exact finding of the splitting transformation is possible only in the simplest cases, the paper proposes and substantiates the possibility of efficient construction of asymptotic expansions of transformation coefficients by powers of small parameters. The stability properties of the integral manifold of slow variables are investigated and the principle of reduction is established, which consists in the fact that the solution of the initial singularly perturbed system will be stable (asymptotically stable, unstable) if and only if the solution of the subsystem of slow variables is stable (asymptotically stable, unstable).