doi: 10.3934/cpaa.2021107
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Applied Analysis, 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[15]

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

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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,  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[15]

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

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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,  Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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) $
[1]

Luis Barreira, Claudia Valls. Reversibility and equivariance in center manifolds of nonautonomous dynamics. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 677-699. doi: 10.3934/dcds.2007.18.677

[2]

Jackson Itikawa, Jaume Llibre, Ana Cristina Mereu, Regilene Oliveira. Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3259-3272. doi: 10.3934/dcdsb.2017136

[3]

P. Yu, M. Han. Twelve limit cycles in a cubic order planar system with $Z_2$- symmetry. Communications on Pure & Applied Analysis, 2004, 3 (3) : 515-526. doi: 10.3934/cpaa.2004.3.515

[4]

Jaume Llibre, Ana Rodrigues. On the limit cycles of the Floquet differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1129-1136. doi: 10.3934/dcdsb.2014.19.1129

[5]

Jaume Llibre, Claudia Valls. Rational limit cycles of Abel equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1077-1089. doi: 10.3934/cpaa.2021007

[6]

Magdalena Caubergh, Freddy Dumortier, Stijn Luca. Cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Lienard systems. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 963-980. doi: 10.3934/dcds.2010.27.963

[7]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[8]

Yunming Zhou, Desheng Shang, Tonghua Zhang. Seventeen limit cycles bifurcations of a fifth system. Conference Publications, 2007, 2007 (Special) : 1070-1081. doi: 10.3934/proc.2007.2007.1070

[9]

Jaume Llibre, Dana Schlomiuk. On the limit cycles bifurcating from an ellipse of a quadratic center. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1091-1102. doi: 10.3934/dcds.2015.35.1091

[10]

José Luis Bravo, Manuel Fernández, Ignacio Ojeda, Fernando Sánchez. Uniqueness of limit cycles for quadratic vector fields. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 483-502. doi: 10.3934/dcds.2019020

[11]

Jaume Llibre, Claudia Valls. Algebraic limit cycles for quadratic polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2475-2485. doi: 10.3934/dcdsb.2018070

[12]

Maoan Han, Tonghua Zhang. Some bifurcation methods of finding limit cycles. Mathematical Biosciences & Engineering, 2006, 3 (1) : 67-77. doi: 10.3934/mbe.2006.3.67

[13]

Zhanyuan Hou, Stephen Baigent. Heteroclinic limit cycles in competitive Kolmogorov systems. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4071-4093. doi: 10.3934/dcds.2013.33.4071

[14]

Maoan Han. On some properties and limit cycles of Lienard systems. Conference Publications, 2001, 2001 (Special) : 426-434. doi: 10.3934/proc.2001.2001.426

[15]

Maoan Han, Yuhai Wu, Ping Bi. A new cubic system having eleven limit cycles. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 675-686. doi: 10.3934/dcds.2005.12.675

[16]

José Luis Bravo, Manuel Fernández, Armengol Gasull. Stability of singular limit cycles for Abel equations. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1873-1890. doi: 10.3934/dcds.2015.35.1873

[17]

Nikolay Dimitrov. An example of rapid evolution of complex limit cycles. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 709-735. doi: 10.3934/dcds.2011.31.709

[18]

Min Li, Maoan Han. On the number of limit cycles of a quartic polynomial system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3167-3181. doi: 10.3934/dcdss.2020337

[19]

Matteo Negri. Crack propagation by a regularization of the principle of local symmetry. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 147-165. doi: 10.3934/dcdss.2013.6.147

[20]

Jon Aaronson, Dalia Terhesiu. Local limit theorems for suspended semiflows. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6575-6609. doi: 10.3934/dcds.2020294

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (37)
  • HTML views (146)
  • Cited by (0)

[Back to Top]