By S. Yakovenko IU S. Iliashenko

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The difference between constant and nonconstant linearization matrices is rather essential in what concerns the size of the common domain of analyticity of all Taylor coefficients of the normal form and/or conjugating transformation. , the field is analytic in D × U , where D is a small polydisk). , the terms that do not commute with the linear field A∗ which is independent of the parameters). All coefficients of all series (the normal form and the conjugacy) will be holomorphic in the maximal natural domain U .

3. Formal flows and embedding 29 To prove that the integral representation gives the same answer as before, it is sufficient to verify it only for the diagonal matrices, when the inverse can be computed explicitly. The advantage of this formula is the possibility of bounding the norm | ln M | defined by the above integral, in terms of |M | and |M −1 |. 13. The matrix logarithm is by no means unique. In the first proof one has the freedom to choose branches of logarithms of eigenvalues arbitrarily and independently for different Jordan blocks.

Note that the leaves are naturally oriented by their complex structure, so the loop generating the local fundamental group is uniquely defined modulo free homotopy. The rest of this section consists of examples, some of them very important for future applications. 21. Consider first the singular foliation spanned by a diagonal linear system x˙ = Ax, A = diag{λ1 , . . 8) This foliation has an isolated singularity (of codimension n) at the origin, and all coordinate axes are complex separatrices.

### Lectures on analytic differential equations by S. Yakovenko IU S. Iliashenko

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