The 16th Hilbert problem restricted to circular algebraic limit cycles

Jaume Llibre; Rafael Ramírez; Valentín Ramírez; Natalia Sadovskaia

doi:https://doi.org/10.1016/j.jde.2015.12.019

The 16th Hilbert problem restricted to circular algebraic limit cycles

Jaume Llibre, Rafael Ramírez, Valentín Ramírez, Natalia Sadovskaia

Department of Mathematics

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

12 Citations (Scopus)

Abstract

© 2015 Elsevier Inc. We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S-1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S-1 algebraic limit cycles given by circles, and this number is reached.

Original language	English
Pages (from-to)	5726-5760
Journal	Journal of Differential Equations
Volume	260
Issue number	7
DOIs	https://doi.org/10.1016/j.jde.2015.12.019
Publication status	Published - 5 Apr 2016

Keywords

16th Hilbert's problem
Algebraic limit circles
Darboux integrability
Invariant algebraic circles
Planar polynomial differential system
Polynomial vector fields

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title = "The 16th Hilbert problem restricted to circular algebraic limit cycles",

abstract = "{\textcopyright} 2015 Elsevier Inc. We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S-1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S-1 algebraic limit cycles given by circles, and this number is reached.",

keywords = "16th Hilbert's problem, Algebraic limit circles, Darboux integrability, Invariant algebraic circles, Planar polynomial differential system, Polynomial vector fields",

author = "Jaume Llibre and Rafael Ram{\'i}rez and Valent{\'i}n Ram{\'i}rez and Natalia Sadovskaia",

year = "2016",

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T1 - The 16th Hilbert problem restricted to circular algebraic limit cycles

AU - Llibre, Jaume

AU - Ramírez, Rafael

AU - Ramírez, Valentín

AU - Sadovskaia, Natalia

PY - 2016/4/5

Y1 - 2016/4/5

N2 - © 2015 Elsevier Inc. We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S-1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S-1 algebraic limit cycles given by circles, and this number is reached.

AB - © 2015 Elsevier Inc. We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S-1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S-1 algebraic limit cycles given by circles, and this number is reached.

KW - 16th Hilbert's problem

KW - Algebraic limit circles

KW - Darboux integrability

KW - Invariant algebraic circles

KW - Planar polynomial differential system

KW - Polynomial vector fields

UR - https://ddd.uab.cat/record/169450

U2 - https://doi.org/10.1016/j.jde.2015.12.019

DO - https://doi.org/10.1016/j.jde.2015.12.019

M3 - Article

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SP - 5726

EP - 5760

IS - 7

ER -