Families of vector fields with many numerical invariants

<p style='text-indent:20px;'>We study bifurcations in finite-parameter families of vector fields on <inline-formula><tex-math id="M1">\begin{document}$S^2$\end{document}</tex-math></inline-formula>. Recently, Yu. Ilyashenko, Kudryashov, and I. Schurov provided examples (locally generic) structurally unstable id="M2">\begin{document}$3$\end{document}</tex-math></inline-formula>-parameter fields: topological classification these admits at least one numerical invariant. They also id="M3">\begin{document}$(2D+1)$\end{document}</tex-math></inline-formula>-parameter such that the has id="M4">\begin{document}$D$\end{document}</tex-math></inline-formula> invariants used those to construct with functional classification.</p><p style='text-indent:20px;'>In this paper, we locally generic id="M5">\begin{document}$4$\end{document}</tex-math></inline-formula>-parameter any prescribed number use them id="M6">\begin{document}$5$\end{document}</tex-math></inline-formula>-parameter invariants. We describe a class id="M7">\begin{document}$3$\end{document}</tex-math></inline-formula>-parameter tail an infinite sequence as invariant classification.</p>