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Orbitally symmetric systems with applications to planar centers

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    * Corresponding author
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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  • 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. $

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

    Citation:

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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

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

    Figure 3.  Phase portrait of vector field 5.7

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

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

    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

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

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

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

    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) $
     | Show Table
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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) $
     | Show Table
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