CENTERS FOR THE KUKLES HOMOGENEOUS SYSTEMS WITH EVEN DEGREE | Zendy

Jaume Llibre | Zendy; Clàudìa Valls | Zendy

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CENTERS FOR THE KUKLES HOMOGENEOUS SYSTEMS WITH EVEN DEGREE

Author(s) -

Jaume Llibre,

Clàudìa Valls

Publication year - 2017

Publication title -

journal of applied analysis and computation

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.55

H-Index - 21

eISSN - 2158-5644

pISSN - 2156-907X

DOI - 10.11948/2017093

Subject(s) - center (category theory) , degree (music) , mathematics , homogeneous , polynomial , combinatorics , zero (linguistics) , mathematical analysis , physics , chemistry , acoustics , crystallography , linguistics , philosophy

For the polynomial differential system x˙ = −y, y˙ = x+Qn(x; y), where Qn(x; y) is a homogeneous polynomial of degree n there are the following two conjectures done in 1999. (1) Is it true that the previous system for n ≥ 2 has a center at the origin if and only if its vector field is symmetric about one of the coordinate axes? (2) Is it true that the origin is an isochronous center of the previous system with the exception of the linear center only if the system has even degree? We prove both conjectures for all n odd.

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