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October  2021, 20(10): 3319-3346. doi: 10.3934/cpaa.2021107

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

Received  July 2020 Revised  May 2021 Published  October 2021 Early access  June 2021

Fund Project: 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).

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. $

Citation: Jefferson L. R. Bastos, Claudio A. Buzzi, Joan Torregrosa. Orbitally symmetric systems with applications to planar centers. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3319-3346. doi: 10.3934/cpaa.2021107
References:
[1]

A. AlgabaC. García and M. Reyes, Quasi-homogeneous linearization of degenerate vector fieldss, J. Math. Anal. Appl., 483 (2020), 123635, 15.  doi: 10.1016/j.jmaa.2019.123635.

[2]

A. Algaba, C. García and J. Giné, Orbital reversibility of planar vector fields, Mathematics, 9, URL https://www.mdpi.com/2227-7390/9/1/14.

[3]

V. I. Arnol'd, Reversible systems, in Nonlinear and turbulent processes in physics, Vol. 3 (Kiev, 1983), Harwood Academic Publ., Chur, 1984, 1161-1174.

[4]

V. I. Arnol'd and M. Sevryuk, Oscillations and bifurcations in reversible systems, in Nonlinear Phenomena in Plasma Physics and Hydrodynamics (ed. R. Sagdeev), Mir, Moscow, 1986, 31-64. doi: 10.1007/BFb0075877.

[5]

N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, American Math. Soc. Translation, 1954 (1954), 19-379. 

[6]

G. D. Birkhoff, The restricted problem of three bodies, Rend. Circ. Mat. Palermo, 39 (1915), 265-334. 

[7]

G. D. Birkhoff, On the periodic motions of dynamical systems, Acta Math., 50 (1927), 359-379.  doi: 10.1007/BF02421325.

[8]

C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc., 312 (1989), 433-486.  doi: 10.2307/2000999.

[9]

C. Chicone and M. Jacobs, Bifurcation of limit cycles from quadratic isochrones, J. Differential Equations, 91 (1991), 268-326.  doi: 10.1016/0022-0396(91)90142-V.

[10]

C. Christopher, Estimating limit cycle bifurcations from centers, in Differential Equations with Symbolic Computation, Trends Math., Birkhäuser, Basel, 2005. doi: 10.1007/3-7643-7429-2_2.

[11]

C. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2007.

[12]

C. Christopher and D. Schlomiuk, On general algebraic mechanisms for producing centers in polynomial differential systems, J. Fixed Point Theory Appl., 3 (2008), 331-351.  doi: 10.1007/s11784-008-0077-2.

[13]

L. V. DetcheniaA. P. Sadovski and T. V. Shcheglova, Reversible cubic systems. Ⅰ, Vesnik of Yank Kupala State University of Grodno, 186 (2015), 13-27. 

[14]

L. V. DetcheniaA. P. Sadovski and T. V. Shcheglova, Reversible cubic systems. Ⅱ, Vesnik of Yank Kupala State University of Grodno, 192 (2015), 13-26. 

[15]

R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113.  doi: 10.2307/1997429.

[16] F. DumortierJ. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006. 
[17]

J. Giné, Higher order limit cycle bifurcations from non-degenerate centers, Appl. Math. Comput., 218 (2012), 8853-8860.  doi: 10.1016/j.amc.2012.02.044.

[18]

J. GinéL. F. S. Gouveia and J. Torregrosa, Lower bounds for the local cyclicity for families of centers, J. Differ. Equ., 275 (2021), 309-331.  doi: 10.1016/j.jde.2020.11.035.

[19]

J. Giné and S. Maza, The reversibility and the center problem, Nonlinear Anal., 74 (2011), 695-704.  doi: 10.1016/j.na.2010.09.028.

[20]

L. F. Gouveia and J. Torregrosa, The local cyclicity problem. Melnikov method using Lyapunov constants, 2020, Preprint.

[21]

L. F. Gouveia and J. Torregrosa, Lower bounds for the local cyclicity of centers using high order developments and parallelization, J. Differ. Equ., 271 (2021), 447-479.  doi: 10.1016/j.jde.2020.08.027.

[22]

M. Han, Bifurcation Theory of Limit Cycles, Science Press Beijing, Beijing; Alpha Science International Ltd., Oxford, 2017.

[23] M. Han and P. Yu, Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles, vol. 181 of Applied Mathematical Sciences, Springer, London, 2012.  doi: 10.1007/978-1-4471-2918-9.
[24]

J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: a survey, Phys. D, 112 (1998), 1-39.  doi: 10.1016/S0167-2789(97)00199-1.

[25]

J. S. W. Lamb and M. Roberts, Reversible equivariant linear systems, J. Differ. Equ., 159 (1999), 239-279.  doi: 10.1006/jdeq.1999.3632.

[26]

Z. Leśniak and Y. G. Shi, One class of planar rational involutions, Nonlinear Anal., 74 (2011), 6097-6104.  doi: 10.1016/j.na.2011.05.088.

[27]

J. LlibreC. Pantazi and S. Walcher, First integrals of local analytic differential systems, Bull. Sci. Math., 136 (2012), 342-359.  doi: 10.1016/j.bulsci.2011.10.003.

[28]

J. Llibre and X. Zhang, On the Darboux integrability of polynomial differential systems, Qual. Theory Dyn. Syst., 11 (2012), 129-144.  doi: 10.1007/s12346-011-0053-x.

[29] D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience Publishers, New York-London, 1955. 
[30]

V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser Boston, Inc., Boston, MA, (2009).  doi: 10.1007/978-0-8176-4727-8.

[31]

R. Roussarie, Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem, Modern Birkhäuser Classics, Birkhäuser/Springer, Basel, 1998. doi: 10.1007/978-3-0348-8798-4.

[32]

M. A. Teixeira, Local reversibility and applications, in Real and complex singularities (São Carlos, 1998), vol. 412 of Chapman & Hall/CRC Res. Notes Math.., Chapman & Hall/CRC, Boca Raton, FL, 2000,

[33]

M. A. Teixeira, Singularities of reversible vector fields, Phys.D, 100 (1997), 101-118.  doi: 10.1016/S0167-2789(96)00183-2.

[34]

A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, vol. 190 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2000. doi: 10.1007/978-3-0348-8440-2.

[35]

H. C. G. von Bothmer, Experimental results for the Poincarécenter problem, Nonlinear Differ. Equ. Appl., 14 (2007), 671-698.  doi: 10.1007/s00030-007-5036-x.

[36]

L. Wei, V. Romanovski and X. Zhang, Generalized involutive symmetry and its application in integrability of differential systems, Z. Angew. Math. Phys., 68 (2017), Paper No. 132, 21. doi: 10.1007/s00033-017-0880-y.

[37]

Y. Zare, Pull Back of Polynomial Differential Equations, PhD thesis, IMPA, Rio de Janeiro, 2017. doi: 10.1090/tran/7660.

[38]

Y. Zare, Center conditions: pull-back of differential equations, Trans. Amer. Math. Soc., 372 (2019), 3167-3189.  doi: 10.1090/tran/7660.

[39]

X. Zhang, Integrability of Dynamical Systems: Algebra and Analysis, vol. 47 of Developments in Mathematics, Springer, Singapore, 2017. doi: 10.1007/978-981-10-4226-3.

[40]

H. Żoładek, The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal., 4 (1994), 79-136.  doi: 10.12775/TMNA.1994.024.

[41]

H. Żołądek, Remarks on:"The classification of reversible cubic systems with center", Topol. Methods Nonlinear Anal., 8 (1996), 335-342.  doi: 10.12775/TMNA.1996.037.

show all references

References:
[1]

A. AlgabaC. García and M. Reyes, Quasi-homogeneous linearization of degenerate vector fieldss, J. Math. Anal. Appl., 483 (2020), 123635, 15.  doi: 10.1016/j.jmaa.2019.123635.

[2]

A. Algaba, C. García and J. Giné, Orbital reversibility of planar vector fields, Mathematics, 9, URL https://www.mdpi.com/2227-7390/9/1/14.

[3]

V. I. Arnol'd, Reversible systems, in Nonlinear and turbulent processes in physics, Vol. 3 (Kiev, 1983), Harwood Academic Publ., Chur, 1984, 1161-1174.

[4]

V. I. Arnol'd and M. Sevryuk, Oscillations and bifurcations in reversible systems, in Nonlinear Phenomena in Plasma Physics and Hydrodynamics (ed. R. Sagdeev), Mir, Moscow, 1986, 31-64. doi: 10.1007/BFb0075877.

[5]

N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, American Math. Soc. Translation, 1954 (1954), 19-379. 

[6]

G. D. Birkhoff, The restricted problem of three bodies, Rend. Circ. Mat. Palermo, 39 (1915), 265-334. 

[7]

G. D. Birkhoff, On the periodic motions of dynamical systems, Acta Math., 50 (1927), 359-379.  doi: 10.1007/BF02421325.

[8]

C. Chicone and M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc., 312 (1989), 433-486.  doi: 10.2307/2000999.

[9]

C. Chicone and M. Jacobs, Bifurcation of limit cycles from quadratic isochrones, J. Differential Equations, 91 (1991), 268-326.  doi: 10.1016/0022-0396(91)90142-V.

[10]

C. Christopher, Estimating limit cycle bifurcations from centers, in Differential Equations with Symbolic Computation, Trends Math., Birkhäuser, Basel, 2005. doi: 10.1007/3-7643-7429-2_2.

[11]

C. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2007.

[12]

C. Christopher and D. Schlomiuk, On general algebraic mechanisms for producing centers in polynomial differential systems, J. Fixed Point Theory Appl., 3 (2008), 331-351.  doi: 10.1007/s11784-008-0077-2.

[13]

L. V. DetcheniaA. P. Sadovski and T. V. Shcheglova, Reversible cubic systems. Ⅰ, Vesnik of Yank Kupala State University of Grodno, 186 (2015), 13-27. 

[14]

L. V. DetcheniaA. P. Sadovski and T. V. Shcheglova, Reversible cubic systems. Ⅱ, Vesnik of Yank Kupala State University of Grodno, 192 (2015), 13-26. 

[15]

R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113.  doi: 10.2307/1997429.

[16] F. DumortierJ. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006. 
[17]

J. Giné, Higher order limit cycle bifurcations from non-degenerate centers, Appl. Math. Comput., 218 (2012), 8853-8860.  doi: 10.1016/j.amc.2012.02.044.

[18]

J. GinéL. F. S. Gouveia and J. Torregrosa, Lower bounds for the local cyclicity for families of centers, J. Differ. Equ., 275 (2021), 309-331.  doi: 10.1016/j.jde.2020.11.035.

[19]

J. Giné and S. Maza, The reversibility and the center problem, Nonlinear Anal., 74 (2011), 695-704.  doi: 10.1016/j.na.2010.09.028.

[20]

L. F. Gouveia and J. Torregrosa, The local cyclicity problem. Melnikov method using Lyapunov constants, 2020, Preprint.

[21]

L. F. Gouveia and J. Torregrosa, Lower bounds for the local cyclicity of centers using high order developments and parallelization, J. Differ. Equ., 271 (2021), 447-479.  doi: 10.1016/j.jde.2020.08.027.

[22]

M. Han, Bifurcation Theory of Limit Cycles, Science Press Beijing, Beijing; Alpha Science International Ltd., Oxford, 2017.

[23] M. Han and P. Yu, Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles, vol. 181 of Applied Mathematical Sciences, Springer, London, 2012.  doi: 10.1007/978-1-4471-2918-9.
[24]

J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: a survey, Phys. D, 112 (1998), 1-39.  doi: 10.1016/S0167-2789(97)00199-1.

[25]

J. S. W. Lamb and M. Roberts, Reversible equivariant linear systems, J. Differ. Equ., 159 (1999), 239-279.  doi: 10.1006/jdeq.1999.3632.

[26]

Z. Leśniak and Y. G. Shi, One class of planar rational involutions, Nonlinear Anal., 74 (2011), 6097-6104.  doi: 10.1016/j.na.2011.05.088.

[27]

J. LlibreC. Pantazi and S. Walcher, First integrals of local analytic differential systems, Bull. Sci. Math., 136 (2012), 342-359.  doi: 10.1016/j.bulsci.2011.10.003.

[28]

J. Llibre and X. Zhang, On the Darboux integrability of polynomial differential systems, Qual. Theory Dyn. Syst., 11 (2012), 129-144.  doi: 10.1007/s12346-011-0053-x.

[29] D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience Publishers, New York-London, 1955. 
[30]

V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser Boston, Inc., Boston, MA, (2009).  doi: 10.1007/978-0-8176-4727-8.

[31]

R. Roussarie, Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem, Modern Birkhäuser Classics, Birkhäuser/Springer, Basel, 1998. doi: 10.1007/978-3-0348-8798-4.

[32]

M. A. Teixeira, Local reversibility and applications, in Real and complex singularities (São Carlos, 1998), vol. 412 of Chapman & Hall/CRC Res. Notes Math.., Chapman & Hall/CRC, Boca Raton, FL, 2000,

[33]

M. A. Teixeira, Singularities of reversible vector fields, Phys.D, 100 (1997), 101-118.  doi: 10.1016/S0167-2789(96)00183-2.

[34]

A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, vol. 190 of Progress in Mathematics, Birkhäuser Verlag, Basel, 2000. doi: 10.1007/978-3-0348-8440-2.

[35]

H. C. G. von Bothmer, Experimental results for the Poincarécenter problem, Nonlinear Differ. Equ. Appl., 14 (2007), 671-698.  doi: 10.1007/s00030-007-5036-x.

[36]

L. Wei, V. Romanovski and X. Zhang, Generalized involutive symmetry and its application in integrability of differential systems, Z. Angew. Math. Phys., 68 (2017), Paper No. 132, 21. doi: 10.1007/s00033-017-0880-y.

[37]

Y. Zare, Pull Back of Polynomial Differential Equations, PhD thesis, IMPA, Rio de Janeiro, 2017. doi: 10.1090/tran/7660.

[38]

Y. Zare, Center conditions: pull-back of differential equations, Trans. Amer. Math. Soc., 372 (2019), 3167-3189.  doi: 10.1090/tran/7660.

[39]

X. Zhang, Integrability of Dynamical Systems: Algebra and Analysis, vol. 47 of Developments in Mathematics, Springer, Singapore, 2017. doi: 10.1007/978-981-10-4226-3.

[40]

H. Żoładek, The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal., 4 (1994), 79-136.  doi: 10.12775/TMNA.1994.024.

[41]

H. Żołądek, Remarks on:"The classification of reversible cubic systems with center", Topol. Methods Nonlinear Anal., 8 (1996), 335-342.  doi: 10.12775/TMNA.1996.037.

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) $
$ 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) $
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) $
$ 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) $
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) $
$ 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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