Review of Research on the Qualitative Theory of Differential Equations at St. Petersburg University. II. Locally Qualitative Analysis of Essentially Nonlinear Systems

This paper is the second in a series of papers devoted to 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 present. The first paper 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 the locally qualitative analysis of essentially nonlinear systems in the neighbourhood of the zero solution, obtained by department staff and graduates. 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 pattern of solution behavior 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 Lozinskii logarithmic 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.