The center problem and composition condition for Abel differential equations

Giné, Jaume; Grau, Maite; Santallusia, Xavier

doi:10.1016/j.exmath.2015.12.002

Cited by 10 publications

(9 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, m, k ≥ 0 are satisfied. It is known for polynomial v i , however, that the moment conditions do not imply the composition condition [22]. The following theorem indicates a condition under which the two conditions are satisfied with respect to the u i functions.…”

Section: Multivariable Hopf Algebra For Toeplitz Multiplicative Outpumentioning

confidence: 99%

See 1 more Smart Citation

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

Ebrahimi‐Fard

Gray

2018

Computation and Combinatorics in Dynamics, Stochastics and Control

View full text Add to dashboard Cite

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 1989, 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.

show abstract

Section: Multivariable Hopf Algebra For Toeplitz Multiplicative Outpumentioning

confidence: 99%

“…It is still believed, however, to be a necessary condition when the v i are polynomials. This is now called the composition conjecture (see [2,5,6,7,47] and the references in the survey article [22]).…”

Section: W Steven Graymentioning

confidence: 99%

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

Ebrahimi‐Fard

Gray

2018

Computation and Combinatorics in Dynamics, Stochastics and Control

View full text Add to dashboard Cite

show abstract

“…It turns out that all the known polynomial Abel differential equations which have a center in [a, b] satisfy the composition condition. Hence in several works was established what is know as composition conjecture, see [2,19] and references therein. This conjecture says that the sufficient condition given in Theorem 1 is also necessary.…”

Section: Introductionmentioning

confidence: 99%

“…To see that the composition condition implies that equation (1) has a center in [a, b] one can consider the transformation y(x) = Y (w(x)) in equation ( 1) in order to obtain the following Abel differential equation It turns out that all the known polynomial Abel differential equations which have a center in [a, b] satisfy the composition condition. Hence in several works was established what is know as composition conjecture, see [2,19] and references therein. This conjecture says that the sufficient condition given in Theorem 1 is also necessary.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A counterexample to the composition condition conjecture for polynomial Abel differential equations

Giné

Grau

Santallusia

2018

Ergod. Th. Dynam. Sys.

Self Cite

View full text Add to dashboard Cite

The Polynomial Abel differential equations are considered a model problem for the classical Poincaré center-focus problem for planar polynomial systems of ordinary differential equations. Last decades several works pointed out that all the centers of the polynomial Abel differential equations satisfied the composition conditions (also called universal centers). In this work we provide a simple counterexample to this conjecture.2010 Mathematics Subject Classification. Primary 34C25. Secondary 34C07.

show abstract

Computation and Combinatorics in Dynamics, Stochastics and Control

2018

Abel Symposia

View full text Add to dashboard Cite

the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

show abstract

The center problem and composition condition for Abel differential equations

Cited by 10 publications

References 24 publications

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

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

A counterexample to the composition condition conjecture for polynomial Abel differential equations

Computation and Combinatorics in Dynamics, Stochastics and Control

Contact Info

Product

Resources

About