On the structure of the space of homogeneous polynomial differential equations of a circle

V. Sh. Roitenberg

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 12 (2020) no. 2, pp. 21-30

Résumé

In this paper, differential equations with their right-hand sides as homogeneous trigonometric polynomials of $n$ degree are examined. The phase space of such equations is a circle. Rough equations, for which the topological structure of the phase portrait does not change when considering a close equation, are described. An equation can be seen as rough if and only if its right-hand side has only simple zeros, that is, all the singular points of which are hyperbolic. The set of all the rough equations is open and is everywhere dense in the space $E_{h}(n)$ of the equations under consideration. The connected components of this set are described. Two rough equations with singular points are attributed to the same component if and only if they are topologically equivalent. In the set of all the non-rough equations, an open and everywhere dense subset is selected, consisting of the equations of the first degree of non-roughness; these are the equations, for which the topological structure of the phase portrait does not change when considering a close non-rough equation. It is an analytic submanifold of codimension one in $E_{h}(n)$ (the bifurcation manifold) and consists of the equations, for which all the singular points are hyperbolic, with the exception of two saddle-node singular points. It is proved that any two rough equations can be connected in $E_{h}(n)$ by a smooth arc with a finite number of bifurcation points where this arc is transversal to the bifurcation manifold.

Détail

Keywords: differential equation on a circle, trigonometric polynomial, roughness, bifurcation manifold, connected component.

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