Homotopy groups of connected components of the set of rough homogeneous polynomial differential equations on the circle

The article considers differential equations on the circle, the right-hand sides of which are homogeneous trigonometric polynomials of degree n. The set E of such equations is identified with the numerical space of ordered sets of coefficients of the corresponding trigonometric polynomials. An equation from E is called rough if the topological structure of its phase portrait does not change when passing to a close equation. The set of rough equations is open and everywhere dense in the space E. We have described the connected components of the set of rough equations and computed their homotopy groups. For connected components containing equations with singular points, the fundamental group is isomorphic to the group Z of integer, and the remaining homotopy groups are zero. For even n, there are also two connected components consisting of equations without singular points. These components are contractible.