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The Faà di Bruno Hopf Algebra for Multivariable Feedback Recursions in the Center Problem for Higher Order Abel Equations

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Book cover Computation and Combinatorics in Dynamics, Stochastics and Control (Abelsymposium 2016)

Part of the book series: Abel Symposia ((ABEL,volume 13))

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Abstract

Poincaré’s center problem asks for conditions under which a planar polynomial system of ordinary differential equations has a center. It is well understood that the Abel equation naturally describes the problem in a convenient coordinate system. In 1990, Devlin described an algebraic approach for constructing sufficient conditions for a center using a linear recursion for the generating series of the solution to the Abel equation. Subsequent work by the authors linked this recursion to feedback structures in control theory and combinatorial Hopf algebras, but only for the lowest degree case. The present work introduces what turns out to be the nontrivial multivariable generalization of this connection between the center problem, feedback control, and combinatorial Hopf algebras. Once the picture is completed, it is possible to provide generalizations of some known identities involving the Abel generating series. A linear recursion for the antipode of this new Hopf algebra is also developed using coderivations. Finally, the results are used to further explore what is called the composition condition for the center problem.

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Author information

Authors and Affiliations

  1. Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  2. Old Dominion University, Norfolk, VA, USA

    W. Steven Gray

Authors
  1. Kurusch Ebrahimi-Fard
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  2. W. Steven Gray
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Correspondence to W. Steven Gray .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Elena Celledoni

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Giulia Di Nunno

  3. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  4. Department of Mathematics, University of Bergen, Bergen, Norway

    Hans Zanna Munthe-Kaas

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Ebrahimi-Fard, K., Steven Gray, W. (2018). The Faà di Bruno Hopf Algebra for Multivariable Feedback Recursions in the Center Problem for Higher Order Abel Equations. In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-01593-0_10

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