On the 16th Hilbert problem for algebraic limit cycles

Jaume Llibre; Rafael Ramírez; Natalia Sadovskaia

doi:https://doi.org/10.1016/j.jde.2009.11.023

On the 16th Hilbert problem for algebraic limit cycles

Jaume Llibre, Rafael Ramírez, Natalia Sadovskaia

Department of Mathematics

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

23 Citations (Scopus)

Abstract

For a polynomial planar vector field of degree n ≥ 2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1 + (n - 1) (n - 2) / 2 when n is even, and (n - 1) (n - 2) / 2 when n is odd. Furthermore, these upper bounds are reached. © 2009 Elsevier Inc. All rights reserved.

Original language	English
Pages (from-to)	1401-1409
Journal	Journal of Differential Equations
Volume	248
DOIs	https://doi.org/10.1016/j.jde.2009.11.023
Publication status	Published - 15 Mar 2010

Keywords

16th Hilbert problem
Algebraic limit cycles
Limit cycles
Polynomial vector fields

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abstract = "For a polynomial planar vector field of degree n ≥ 2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1 + (n - 1) (n - 2) / 2 when n is even, and (n - 1) (n - 2) / 2 when n is odd. Furthermore, these upper bounds are reached. {\textcopyright} 2009 Elsevier Inc. All rights reserved.",

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AU - Llibre, Jaume

AU - Ramírez, Rafael

AU - Sadovskaia, Natalia

PY - 2010/3/15

Y1 - 2010/3/15

N2 - For a polynomial planar vector field of degree n ≥ 2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1 + (n - 1) (n - 2) / 2 when n is even, and (n - 1) (n - 2) / 2 when n is odd. Furthermore, these upper bounds are reached. © 2009 Elsevier Inc. All rights reserved.

AB - For a polynomial planar vector field of degree n ≥ 2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1 + (n - 1) (n - 2) / 2 when n is even, and (n - 1) (n - 2) / 2 when n is odd. Furthermore, these upper bounds are reached. © 2009 Elsevier Inc. All rights reserved.

KW - 16th Hilbert problem

KW - Algebraic limit cycles

KW - Limit cycles

KW - Polynomial vector fields

U2 - https://doi.org/10.1016/j.jde.2009.11.023

DO - https://doi.org/10.1016/j.jde.2009.11.023

M3 - Article

VL - 248

SP - 1401

EP - 1409

ER -