Uniqueness of limit cycles for quadratic vector fields

José Luis Bravo; Manuel Fernández; Ignacio Ojeda; Fernando Sánchez

doi:10.3934/dcds.2019020

Article Contents

2019, Volume 39, Issue 1: 483-502. Doi: 10.3934/dcds.2019020

This issue Previous Article The conditional variational principle for maps with the pseudo-orbit tracing property Next Article Non-local sublinear problems: Existence, comparison, and radial symmetry

Uniqueness of limit cycles for quadratic vector fields

Departamento de Matemáticas, Universidad de Extremadura, Badajoz 06006, Spain

* Corresponding author: J. L. Bravo
* Corresponding author: J. L. Bravo

Received: March 31, 2018

Published: December 2018

The first two authors were partially supported by AEI/FEDER UE grant number MTM 2011-22751 and Junta de Extremadura grant GR15055 (Junta de Extremadura/FEDER funds). The third author was partially supported by the research group FQM-024 (Junta de Extremadura/FEDER funds) and by the project MTM2015-65764-C3-1-P (MINECO/FEDER, UE). The fourth author was partially supported by Junta de Extremadura grant GR15055 (Junta de Extremadura/FEDER funds).

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Abstract

This article deals with the study of the number of limit cycles surrounding a critical point of a quadratic planar vector field, which, in normal form, can be written as $x' = a_1 x-y-a_3x^2+(2 a_2+a_5)xy + a_6 y^2$, $y' = x+a_1 y + a_2x^2+(2 a_3+a_4)xy -a_2y^2$. In particular, we study the semi-varieties defined in terms of the parameters $a_1, a_2, ..., a_6$ where some classical criteria for the associated Abel equation apply. The proofs will combine classical ideas with tools from computational algebraic geometry.

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Mathematics Subject Classification: Primary: 34C25; Secondary: 34A34, 37C27, 37G15.

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Table 1. Codimensions of the semi-varieties.

Case	Point	$c_p$	$c_I$
1a)	$a_1=1$, $a_2=0$, $a_3=1$, $a_4=-3$, $a_5=0$, $a_6=0$.	4	4
1b)	$a_1=1$, $a_2=1$, $a_3=-1$, $a_4=2$, $a_5=-4$, $a_6=1$.	4	4
2)	$a_1=-1$, $a_2=\sqrt{14}$, $a_3=-2$, $a_4=-1$, $a_5=-3 \sqrt{14}$, $a_6=0$.	3	3
3a)	$a_1=-1$, $a_2=(201 + 2 \sqrt{1509})/58$, $a_3=(-33 + 4 \sqrt{1509})/58$, $a_4=-1$, $a_5=-16$, $a_6=(-201 - 2 \sqrt{1509})/58)$	3	3
3b)	$a_1=0$, $a_2=0$, $a_3=0$, $a_4=1$, $a_5=-2$, $a_6=-1$.	5	5
4)	$a_1=1$, $a_2=0$, $a_3=1/3$, $a_4=-1$, $a_5=-1$, $a_6=0$.	3	3
5a)	$a_1=0$, $a_2=1$, $a_3=-15/16$, $a_4=-53/16$, $a_5=(-941 - 31 \sqrt{7913})/512$, $a_6=1$.	1	1
5b)	$a_1=0$, $a_2=(4096 - 7 \sqrt{1726})/16384$, $a_3=0$, $a_4=-(58339673 + 28672 \sqrt{1726})/94666752$, $a_5=-1$, $a_6=-2889/16384$.	2	2
5c)	$a_1=1$, $a_2=4$, $a_3=-12$, $a_4=30$, $a_5=-15$, $a_6=1/2$	*	2
5d)	$a_1=0$, $a_2=\sqrt{185}/32$, $a_3=0$, $a_4=-1$, $a_5=-3 \sqrt{185}/32$, $a_6=-5/32$	2	2
5e)	$a_1=0$, $a_2=2 \sqrt{2}$, $a_3=-1$, $a_4=0$, $a_5=-9 \sqrt{2}$, $a_6=8$	*	2
5f)	$a_1=0$, $a_2=2/3$, $a_3=0$, $a_4=-1$, $a_5=-2$, $a_6=-1/3$	3	2
5g)	$a_1=0$, $a_2=1$, $a_3=-9/2$, $a_4=15/2$, $a_5=-15$, $a_6=8$	*	2
5h)	$a_1=0$, $a_2=1$, $a_3=-8$, $a_4=35/2$, $a_5=-15$, $a_6=9/2$	*	2

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