Topological types of polynomial differential equations

Abstract:

Consider a first order system of real ordinary differential equations, with polynomial coefficients, having no critical points in the number space ${R^n}$. Two such differential systems are called topologically equivalent in case there exists a homeomorphism of ${R^n}$ onto itself carrying the sensed (not parametrized) solutions of the first system onto the solution family of the second system. Let ${B^n}(m)$ be the cardinal number of topological equivalence classes for systems in ${R^n}$ with polynomial coefficients of degree at most $m$. The author proves that ${B^2}(m)$ is finite and obtains explicit upper and lower bounds in terms of $m$. Also examples are given to show that ${B^n}(m)$ is noncountable for $n \geqslant 3$ and $m \geqslant 6$.