Orbitally symmetric systems with applications to planar centers

Jefferson L. R. Bastos; Claudio A. Buzzi; Joan Torregrosa

doi:10.3934/cpaa.2021107

Article Contents

2021, Volume 20, Issue 10: 3319-3346. Doi: 10.3934/cpaa.2021107 open access

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Orbitally symmetric systems with applications to planar centers

1.
Departamento de Matemática, Universidade Estadual Paulista "Júlio de Mesquita Filho", 15054-000 São José do Rio Preto, Brazil
2.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
3.
Centre de Recerca Matemàtica, Campus de Bellaterra, 08193 Bellaterra, Barcelona, Spain

^* Corresponding author
^* Corresponding author

Received: June 30, 2020

Revised: April 30, 2021

Early access: June 2021

Published: October 2021

This work has received funding from the Ministerio de Economía, Industria y Competitividad Agencia Estatal de Investigación (MTM2016-77278-P FEDER and PID2019-104658GBI00 grants), the Agència de Gestió d'Ajuts Universitaris i de Recerca (2017 SGR 1617 grant), the Brazilian agencies FAPESP (2013/24541-0, 2017/03352-6 and 2019/10269-3 grants), CAPES (88881.068462/2014-01 PROCAD grant), CNPq (308006/2015-1 grant), and the European Union's Horizon 2020 research and innovation programme (Dynamics-H2020-MSCA-RISE-2017-777911 grant).

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Abstract

We present a generalization of the most usual symmetries in differential equations known as the time-reversibility and the equivariance ones. We check that the typical properties are also valid for the new definition that unifies both. With it, we are able to present new families of planar polynomial vector fields having equilibrium points of center type. Moreover, we provide the highest lower bound for the local cyclicity of an equilibrium point of polynomial vector fields of degree 6, $ M(6)\ge 48. $

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Mathematics Subject Classification: Primary: 34C14, 34C25, 34C07.

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Figure 1. Phase portraits of systems 4.1 and 4.2. The fixed points set of the involution are depicted in red

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Figure 2. Phase portraits of 5.1 for $ \lambda $ equal to $ \lambda_a, $ $ \lambda_b, $ and $ \lambda_c $, respectively

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Figure 3. Phase portrait of vector field 5.7

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Figure 4. Phase portrait of 5.8 for $ (a,b,c) = (2,1,1) $ and $ (a,b,c) = (2,1,-1) $

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Figure 5. Phase portrait of 5.9 for $ (a,b) = (2,2) $ and $ (a,b) = (2,-3) $

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Figure 6. Phase portrait of 5.10 for $ k = 13/4 $ and $ k = 2 $ with the respective zooms near the center and the saddle symmetric equilibria

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Figure 7. Phase portrait of 6.1 for $ a = -1,-1/12,1/4,1/2,2 $

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Figure 8. Phase portrait of 6.2 with a zoom near the center point

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Figure 9. Phase portrait of 6.3 with two zooms near the center point

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Table 1. Functions $ \Phi $ and $ \varphi $ corresponding to the reversible families $ CR_m^{(n)}, $ for $ m = 1,\ldots,17, $ in [40]. The functions $ T_1,T_2 $ are defined in 3.2 and $ \alpha,\beta,\gamma, $ and $ \delta $ in 3.3

$ CR_m^{(n)} $	$ \Phi(x,y) $	$ \varphi(x,y) $

$ CR_1^{(7)} $	$ \left(x^2,y\right) $	$ \left(-x,y\right) $
$ CR_{2^\ast}^{(10)} $	$ \left(x,y^2/(h(x)+y)\right) $	$ \left(x,-h(x) y/(h(x)+y)\right) $
$ CR_3^{(10)} $	$ \left(x,\frac{y^2}{x y+a x^2+b x+1}\right) $	$ \left(x, \frac{-y (a x^2+b x+1)}{a x^2+b x+x y+1}\right) $
$ \begin{array}{l}CR_4^{(8)}\;\;CR_5^{(8)}\\CR_6^{(7)}\;\;CR_7^{(9)}\end{array} $	$ \left(T_1 x,T_1/y\right) $	$ \left(-c-x, -x y/(x+c)\right) $
$ CR_{8}^{(10)} $ $ CR_{9}^{(10)} $	$ \left(T_1 x,T_1^2/y\right) $	$ \left(\alpha, x(\alpha+c+x)/\alpha\right) $
$ CR_{10}^{(10)} $	$ \left(T_1 x,T_1^3/y\right) $	$ \left(\beta, -\frac{\beta^2+\beta c-c x-x^2-x y}{\beta}\right) $
$ CR_{11}^{(7)} $ $ CR_{12}^{(7)} $	$ \left(T_1^2 x,T_1/y\right) $	$ \left(c^2/x, yc/x\right) $
$ CR_{13}^{(10)} $ $ CR_{14}^{(9)} $	$ \left(T_1^3/x,T_1^2/y\right) $	$ \left(\frac{c^2 y-\beta(x(x+2y)(c+y) + y^3)}{x (\beta x-c-x-y)}, \beta y \right) $
$ CR_{15}^{(10)} $	$ \left(T_1^4/x,T_1^2/y\right) $	$ \left(\gamma^2 x, \gamma y \right) $
$ CR_{16}^{(5)} $	$ \left(T_2/x,T_2/y\right) $	$ \left( \frac{x}{a x^2+b x y+c y^2}, \frac{ y}{a x^2+b x y+c y^2}\right) $
$ CR_{17}^{(12)} $	$ \left(\frac{x^3}{y},\frac{x^2}{x y - a y^2 + 2 x + 2 (1 + a) y + 1 - a}\right) $	$ \left(\delta x, \delta^3 y \right) $

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Table 2. Rational involutions $ \varphi $ corresponding to new reversible families $ CR_m, $ for $ m = 18,19,20 $

$ CR_m $	$ \varphi(x,y) $

$ CR_{18} $ $ CR_{19} $	$ \left(x,(2x+y-1/(y-1)\right) $
$ CR_{20} $	$ \left(x,(x^2-xy+2x-y)/(x+1)\right) $

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Table 3. Rational diffeomorphisms $ \phi $ and involutions $ \varphi $ corresponding to new reversible families $ CR_m, $ for $ m = 21,22,23 $

$ CR_m $	$ \phi(x,y) $	$ \varphi(x,y) $

$ CR_{21} $	$ \left(x+ay^2, y\right) $	$ \left(-x-2ay^2, y\right) $
$ CR_{22} $	$ \left(x+ay^2+by^3, y\right) $	$ \left(-x-2ay^2-2by^3, y\right) $
$ CR_{23} $	$ \left(x+ay^2/(y^2+1), y\right) $	$ \left(-(2ay^2+xy^2+x)/(y^2+1), y\right) $

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