Review of Research on the Qualitative Theory of Differential Equations at St. Petersburg University. I. Stable Periodic Points of Diffeomorphisms with Homoclinic Points and Systems with Weakly Hyperbolic Invariant Sets

This paper is the first in a series of review publications devoted to the results of scientific research that has been carried out in the Department of Differential Equations of St. Petersburg University over the past 30 years. The current scientific interests of the department staff can be divided into the following directions and topics: studying stable periodic points of diffeomorphisms with homoclinic points, the study of systems with weakly hyperbolic invariant sets, local qualitative theory of essentially nonlinear systems, classification of phase portraits of a family of cubic systems, and the stability conditions for systems with hysteretic nonlinearities and systems with nonlinearities under the generalized Routh–Hurwitz conditions (Aizerman problem). This paper presents recent results on the first two topics outlined above. The study of stable periodic points of diffeomorphisms with homoclinic points is carried out under the assumption that the stable and unstable manifolds of hyperbolic points are tangent at a homoclinic (heteroclinic) point, and the homoclinic (heteroclinic) point is not a point with a finite order of tangency. The research of systems with weakly hyperbolic invariant sets is conducted for the case when neutral, stable, and unstable linear spaces do not satisfy the Lipschitz condition.