Normal forms and linearization of resonant vector fields with multiple eigenvalues (Q705309)

A germ of a nonlinear vector field \(X(x)=Ax+a(x),\) with \(a(x)=O(x^2),\) where \(A\) is a resonant matrix with multiple eigenvalues is considered. The linearization and normal forms of vector field \(X\) are discussed in the formal category. In [\textit{J. Basto-Gonçalves}, Linearization of resonant vector fields, Universite de Bourgogne, Preprint (2003)], the sets \({\mathcal G}\) and \({\mathcal B}\) of resonant monomials are introduced such that if all the resonant monomials in the nonlinearity \(a(x)\) are in \({\mathcal G}\), then the vector field \(X\) is linearizable; a resonant normal form for X can be obtained involving only the nonlinear resonant monomials from \({\mathcal B}\). To find the set \({\mathcal B}\), an oriented graph to the resonant monomials is constructed. A special procedure allows one to simplify the connected components of this graph. The elements of \({\mathcal B}\) are exactly those components remaining in the graph after simplification. The construction of \({\mathcal B}\) allows one to indicate the resonant monomials from which a resonant normal form can be obtained. Corresponding theorems are proved when the matrix \(A\) has only one Jordan block of dimension 2 and 3, or two Jordan blocks, both of dimension 2.