Review of the research on the qualitative theory of differential equations at St. Petersburg University. II. Locally qualitative analysis of essentially nonlinear systems

Abstract

This article is the second one in a series of articles devoted to a review of the results of scientific research that have been carried out at the Department of Differential Equations of St.Petersburg University over the past three decades and continue to be carried out at the present time. The first article talked about studies of stable periodic points of diffeomorphisms with homoclinic points and systems of differential equations with weakly hyperbolic invariant sets. This paper presents the results of locally qualitative analysis of essentially nonlinear systems in the neighbourhood of the zero solution, obtained by employees and graduates of the department. A system is said to be essentially nonlinear if the Taylor series expansion of its right-hand sides does not contain linear terms. The study of such systems, firstly, is complicated by a more complex picture of the behavior of solutions compared to quasilinear systems. Secondly, there are not even theoretical formulas for the general solution of a nonlinear first-approximation system, the presence of which is so helpful in the quasi-linear case. All this complicates analysis and significantly limits technical capabilities. Therefore, almost any new results and any new methods of working with such systems are of great interest. One of the most effective tools for working with essentially nonlinear systems turned out to be the logarithmic Lozinsky norms. In a sense, they are an analogue of the characteristic exponents (eigenvalues) used in the theory of quasilinear systems. Research conducted at the department has demonstrated the wide possibilities of using logarithmic norms in a wide variety of problems.

Keywords:

qualitative theory of differential equations, essentially nonlinear systems, logarithmic Lozinski norms, invariant surfaces, Lyapunov stability, smooth equivalence