First integrals of local analytic differential systems

Jaume Llibre; Chara Pantazi; Sebastian Walcher

doi:https://doi.org/10.1016/j.bulsci.2011.10.003

First integrals of local analytic differential systems

Jaume Llibre, Chara Pantazi, Sebastian Walcher

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

24 Citations (Scopus)

Abstract

We investigate formal and analytic first integrals of local analytic ordinary differential equations near a stationary point. A natural approach is via the Poincaré-Dulac normal forms: If there exists a formal first integral for a system in normal form then it is also a first integral for the semisimple part of the linearization, which may be seen as "conserved" by the normal form. We discuss the maximal setting in which all such first integrals are conserved, and show that all first integrals are conserved for certain classes of reversible systems. Moreover we investigate the case of linearization with zero eigenvalues, and we consider a three-dimensional generalization of the quadratic Dulac-Frommer center problem. © 2011 Elsevier Masson SAS.

Original language	English
Pages (from-to)	342-359
Journal	Bulletin des Sciences Mathematiques
Volume	136
Issue number	3
DOIs	https://doi.org/10.1016/j.bulsci.2011.10.003
Publication status	Published - 1 Apr 2012

Keywords

Center
Formal first integral
Normal form
Primary
Reversible system

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AU - Pantazi, Chara

AU - Walcher, Sebastian

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AB - We investigate formal and analytic first integrals of local analytic ordinary differential equations near a stationary point. A natural approach is via the Poincaré-Dulac normal forms: If there exists a formal first integral for a system in normal form then it is also a first integral for the semisimple part of the linearization, which may be seen as "conserved" by the normal form. We discuss the maximal setting in which all such first integrals are conserved, and show that all first integrals are conserved for certain classes of reversible systems. Moreover we investigate the case of linearization with zero eigenvalues, and we consider a three-dimensional generalization of the quadratic Dulac-Frommer center problem. © 2011 Elsevier Masson SAS.

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KW - Formal first integral

KW - Normal form

KW - Primary

KW - Reversible system

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EP - 359

JO - Bulletin des Sciences Mathematiques

JF - Bulletin des Sciences Mathematiques

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