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Moduli of stability for germs of homogeneous vectorfields on \(R^ 3\). (English) Zbl 0633.58031
It is well known that a finite classification for the homogeneous quadratic vector fields in the plane is possible. The authors construct an example of an uncountable family of homogeneous quadratic vector fields \(X_ a\) in \(R^ 3\). They prove two theorems. Theorem A: If \(X_ a\) is topologically equivalent to \(X_{a'}\), at the origin in \(R^ 3\) then \(a=a'\). Theorem \(B: X_ a\) have a nonstabilizable modulus of stability on the 2-jet level in the following sense. If \(Y_ a\) and \(Y_{a'}\) are vector fields such that \(X_ a\), \(Y_ a\) and \(Y_{a'}\) have the same Taylor expansion at the origin up to order 2, then \(a\neq a'\) implies that \(Y_ a\) and \(Y_{a'}\) are topologically nonequivalent at the origin. The bibliography contains 37 items.
Reviewer: A.Kanevskij

MSC:
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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