The local structure of multiple solutions of a non-linear equation in the case of multidimensional degeneration

Abstract

Background. When studying mathematical models of physical processes described by nonlinear equations with a parameter, situations are identified when the parameter reaches a certain critical value, which leads to a violation of unambiguity in the behavior of the process, which is expressed in the branching of solutions to the equation. Starting from the works of A. M. Lyapunov devoted to the study of the forms of equilibrium figures of a rotating fluid; the solution of the branching problem is reduced to solving a finitedimensional equation, which has received the name of the branching equation. For the case of small dimensions of the branching equation, numerous works solve the problem of constructing the branching equation and its investigation. However, the general theory of branching is not yet complete. In particular, the question is not solved under which conditions small changes in the parameter in the vicinity of the critical value do not change the local topological structure of the branch. This paper is devoted to one of the approaches to solving this problem. Materials and methods. The results of the work are based on the use of methods of the theory of singularities of differentiable maps, developed by R. Thom. Results and conclusions. A class of maps is distinguished, belonging to which makes it possible to reduce the branching equation to a polynomial one. For a local set of solutions of the branching equation, a trivial bundle is constructed that establishes a local homeomorphism of sections of the set of solutions for different values of the parameter. This makes it possible to determine the nature of a physical process in the vicinity of the critical parameter value when studying it.

Keywords

• branching equation
• tom mapping
• commutative diagram
• stratification
• algebraic set