Speciality:
01.01.02 (Differential equations, dynamical systems, and optimal control)
Keywords: linear and nonlinear functional analysis; nonlinear equations; branching theory of solutions of nonlinear equations; stability; symmetry in nonlinear analysis; applications to mathematical physics and natural sciences.
Subject:
In 1964–1969 — works on spectral theory of linear operators (perturbations of discrete spectrum, numerical range and spectrum, von Neumann spectral sets, generalizations of normal operators). From 1969 — works on branching theory of solutions of nonlinear equations, mainly under group symmetry conditions. They were proved theorems about symmetry of basic nonlinear equation inheritance by branching equations (BEq). On their base group analysis of differential equations methods were applied to construction of BEq general form on allowing by its group symmetry both in stationary and nonstationary branching. Role of generalized Jordan structure of the linearized operator-function of bifurcational parameter in branching theory, recently in application to the construction and investigation of resolving systems in branching theory problems, in particular to BEq in the root subspaces. Applications to mathematical physics: phase transitions in statistical crystal theory; capillary-gravity surface waves in fluid layers; von Karman plate in supersonic gas flow; pseudoperturbation methods in spectral theory.
Main publications:
Loginov B. V. Determination of the bifurcation equation by its group symmetry — Andronov–Hopf bifurcation // Nonlinear Analysis, TMA, 28(1997), 12, 2033–2047.
