Periodic differential second order systems, invariance relative to rotations

Background. For the theory of differential equations and its applications, it is of interest to study the structure of typical dynamical systems with symmetry. For two-dimensional autonomous systems invariant under the expansion group as well as the rotation group and its finite subgroups, such studies have already been published. This paper discusses periodic systems of differential equations in the unit disc D on the plane, invariant with respect to the group of rotations. The aim of the paper is to describe an open and everywhere dense set in the space S of such systems. Materials and methods. We use methods of the qualitative theory of differential equations and functional analysis. Results and Conclusions. The canonical form of the considered differential systems in polar coordinates is obtained. The structures of phase portraits of generic systems from S are described. The necessary and sufficient conditions for the roughness of systems with respect to the space S are obtained. It is shown that rough systems are not dense in the space S.