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Poincare's center problem asks for conditions under which a planar polynomialsystem of ordinary differential equations has a center. It is well understoodthat the Abel equation naturally describes the problem in a convenientcoordinate system. In 1989, Devlin described an algebraic approach forconstructing sufficient conditions for a center using a linear recursion forthe generating series of the solution to the Abel equation. Subsequent work bythe authors linked this recursion to feedback structures in control theory andcombinatorial Hopf algebras, but only for the lowest degree case. The presentwork introduces what turns out to be the nontrivial multivariablegeneralization of this connection between the center problem, feedback control,and combinatorial Hopf algebras. Once the picture is completed, it is possibleto provide generalizations of some known identities involving the Abelgenerating series. A linear recursion for the antipode of this new Hopf algebrais also developed using coderivations. Finally, the results are used to furtherexplore what is called the composition condition for the center problem.

The Faa di Bruno Hopf algebra for multivariable feedback recursions in the center problem for higher order Abel equations