The theory of singularities is an important part of various branches of mathematics: algebraic geometry, differential topology, geometric optics, etc. Here the focus is on the singularities of smooth maps and applications to dynamical systems - in particular, bifurcations. This includes the study of bifurcations of intersections of stable and unstable cycles. Along with the formal algebraic and analytic aspects of the theory, the authors consider global topological problems related to invariants. The authors have in mind a student reader, mathematician, or physicist, who wishes to learn the modern techniques of local mathematical analysis as an instrument for applied studies or a specialist in one of the applied areas who is looking for the necessary mathematical tools.