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Unique normal forms for planar vector fields. (English) Zbl 0691.58012

The authors prove and then generalize the following result: Let X be a germ of an analytic vector field in equilibrium at \(0\in R^ 2\) with purely imaginary eigenvalues \(\pm a_ 0i\), \(a_ 0>0\). Then X is formally equivalent to precisely one of the following polynomial vector fields (where \(x=\sqrt{\rho} \cos \phi\), \(y=\sqrt{\rho}\sin\phi\) for \((x,y)\in R^ 2):\) \(a_ 0\partial /\partial \phi\), \((a_0\pm\rho^ k)\partial/\partial\phi\), \[ (a_ 0 + a_1 \rho + \cdots + a_ j \rho^ j)\partial/\partial \phi \pm \rho^ j (1+b\rho^ j)\partial /\partial \rho \quad (j>0). \] A graded Lie algebra \(\mathcal N\) acting as derivations on the power series ring \(K[[u]]\) in one variable where \(K=R\) or C that is free of rank 2 over \(K[[u]]\) and with generators in an Abelian \(\mathcal N_ 0\) is called an \(\mathcal A\mathcal B\)-algebra. Basic properties of \(\mathcal A\mathcal B\)-algebras are studied in the paper. The main theorem concerns planar vector fields and gives the list of possible normal forms in an \(\mathcal A\mathcal B\)-algebra \(\mathcal N\) for the action of \(\mathcal G\times \mathcal H\) where \(\mathcal G=\exp (ad(\mathcal N_+))\) is a group of automorphisms of \(\mathcal N\) and \(\mathcal H\) is either \(R_*\) or \(R_+\) (if \(K=R)\) or \(C_*\) (if \(K=C):\) \(aA_ 0\quad (a\in K),aA_ 0+\sigma A_ k\quad (k>0,\quad a\in K,\quad \sigma \in K_*/{\mathcal H}^ k),aA_ 0+bB_ 0\quad (0\neq b,\quad a\in K),pA_ 0+\sigma B_ j+bB_{2j}\quad (j>0,\quad b\in K,\quad \sigma \in K_*/\mathcal H^ j,\quad p \text{ a polynomial of degree at most }j)\)where \(0\neq A_ 0\), \(B_ 0\in {\mathcal N}_ 0\) such that \(A_ 0(u)=0\), \(B_ 0(u)=u\); \(A_ j=u^ jA_ 0\), \(B_ j=u^ jB_ 0\). The uniqueness of these forms and some special cases are then discussed.
Reviewer: J.Durdil

MSC:

58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
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References:

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