On structural stability and bifurcations of polynomial differential equations on the circle

A dynamical system defined by a differential equation on a manifold, the phase space of a system, is called structurally stable if the topological structure of the phase portrait does not change when passing to a close equation. The concept of structural stability emerged from the idea that the essential properties of a dynamical system describing a real process should not change with small changes in the parameters of the system. By now, natural necessary and sufficient conditions for structural stability of dynamical systems on closed manifolds of any dimension have been obtained. However, if we consider structural stability in narrower classes of dynamical systems, in particular, in the space of systems defined by differential equations with polynomial right-hand sides, the conditions of structural stability have not been studied even for small dimensions of the phase space. This paper considers the dynamic systems given by the differential equations, the right-hand parts of which are trigonometric polynomials of the degree not exceeding the number $n$. The phase space of such systems is a circle. We describe equations that are structurally stable with respect to the space $E(n)$ of all such equations. An equation is structurally stable if and only if its right-hand side has only simple zeros, that is, all singular points of which are hyperbolic. The set of all structurally stable equations is open and is dense everywhere in the space $E(n)$. In the set of all non-structurally-stable equations, an open, everywhere-dense subset consisting of equations that are first order structurally unstable is distinguished. It is an analytic submanifold of codimension one in $E(n)$ and consists of equations for which all the zeros of the right-hand side are simple, except for one double zero.