abstract
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We study bifurcations in finite-parameter families of vector fields on
. Recently, Yu. Ilyashenko, Yu. Kudryashov, and I. Schurov provided examples of (locally generic) structurally unstable\begin{document}$S^2$end{document} -parameter families of vector fields: topological classification of these families admits at least one numerical invariant. They also provided examples of\begin{document}$3$end{document} -parameter families such that the topological classification of these families has at least\begin{document}$(2D+1)$end{document} numerical invariants and used those examples to construct families with functional invariants of topological classification.\begin{document}$D$end{document} In this paper, we construct locally generic
-parameter families with any prescribed number of numerical invariants and use them to construct\begin{document}$4$end{document} -parameter families with functional invariants. We also describe a locally generic class of\begin{document}$5$end{document} -parameter families with a tail of an infinite number sequence as an invariant of topological classification.\begin{document}$3$end{document}