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July  2022, 27(7): 3991-4006. doi: 10.3934/dcdsb.2021215

Isochronicity of bi-centers for symmetric quartic differential systems

Departamento de Matemática e Estatística, Universidade Federal de São João del-Rei, São João del-Rei, 36307-352, Brazil

*Corresponding author: Wilker Fernandes (wilker@ufsj.edu.br)

Received  February 2021 Revised  June 2021 Published  July 2022 Early access  September 2021

Fund Project: The authors thank the reviewer for careful reading and valuable suggestions which helped to improve the manuscript. W. Fernandes and P. Tempesta are partially supported by FAPESP Grand number 2019/07316-0.

In this paper we investigate the simultaneous existence of isochronous centers for a family of quartic polynomial differential systems under four different types of symmetry. Firstly, we find the normal forms for the system under each type of symmetry. Next, the conditions for the existence of isochronous bi-centers are presented. Finally, we study the global phase portraits of the systems possessing isochronous bi-centers. The study shows the existence of seven non topological equivalent global phase portraits, where three of them are exclusive for quartic systems under such conditions.
 
Addendum: "W. Fernandes and P. Tempesta are partially supported by FAPESP Grand number 2019/07316-0." is added under Fund Project. We apologize for any inconvenience this may cause.

Citation: Wilker Fernandes, Viviane Pardini Valério, Patricia Tempesta. Isochronicity of bi-centers for symmetric quartic differential systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3991-4006. doi: 10.3934/dcdsb.2021215
References:
[1]

J. C. Artés, F. Dumortier, C. Herssens, J. Llibre and P. De Maesschalck, Computer program P4 to study phase portraits of plane polynomial differential equation, 2003, Available from: http://mat.uab.es/ artes/p4/p4.htm.

[2]

J. ChavarrigaI. A. García and J. Giné, Isochronicity into a family of time-reversible cubic vector fields, Appl. Math. Comput., 121 (2001), 129-145.  doi: 10.1016/S0096-3003(99)00267-2.

[3]

J. ChavarrigaI. A. García and J. Giné, Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomial, Bull. Sci. Math., 123 (1999), 77-96.  doi: 10.1016/S0007-4497(99)80015-3.

[4]

J. ChavarrigaI. A. García and J. Giné, Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials, J. Comput. Appl. Math., 126 (2000), 351-368.  doi: 10.1016/S0377-0427(99)00364-7.

[5]

L. ChenZ. Lu and D. Wang, A class of cubic systems with two centers or two foci, J. Math. Anal. Appl., 242 (2000), 154-163.  doi: 10.1006/jmaa.1999.6630.

[6]

T. ChenS. Li and J. Llibre, Z$_2$-equivariant linear type bi-center cubic polynomial Hamiltonian vector fields, J. Differ. Equ., 269 (2020), 832-861.  doi: 10.1016/j.jde.2019.12.020.

[7]

X. ChenW. HuangV. G. Romanovski and W. Zhang, Linearizability conditions of a time-reversible quartic-like system, J. Math. Anal. Appl., 383 (2011), 179-189.  doi: 10.1016/j.jmaa.2011.05.018.

[8]

X. ChenV. G. Romanovski and W. Zhang, Linearizability conditions of time-reversible quartic systems having homogeneous nonlinearities, Nonlinear Anal., 69 (2008), 1525-1539.  doi: 10.1016/j.na.2007.07.009.

[9]

A. CimaA. GasullV. Mañosa and F. Mañosas, Algebraic properties of the Liapunov and period constants, Rocky Mountain J. Math., 27 (1997), 471-501.  doi: 10.1216/rmjm/1181071923.

[10]

R. Conti, On a class of cubic systems with two centers, Results Math., 14 (1988), 30-37.  doi: 10.1007/BF03323214.

[11]

W. Decker, G. M. Greuel, G. Pfister and H. A.Shönemann, SINGULAR 4-2-1 –- A Computer algebra system for polynomial computations, 2021, Available from: http://www.singular.uni-kl.de.

[12]

C. Du, The problem of bicenter and isochronicity for a class of quasi symmetric planar systems, Abstr. Appl. Anal., (2014), 482450. doi: 10.1155/2014/482450.

[13]

M. DukarićW. Fernandes and R. Oliveira, Symmetric centers on planar cubic differential systems, Nonlinear Anal., 197 (2020), 111868.  doi: 10.1016/j.na.2020.111868.

[14]

W. FernandesR. Oliveira and V. G. Romanovski, Isochronicity for a $\mathbb Z_2$-equivariant quintic system, J. Math. Anal. Appl., 467 (2018), 874-892.  doi: 10.1016/j.jmaa.2018.07.053.

[15]

W. Fernandes, V. G. Romanovski, M. Sultanova and Y. Tang, Isochronicity and linearizability of a planar cubic system, J. Math. Anal. Appl., 450 (2017), 795–813. doi: 10.1016/j.jmaa.2017.01.058.

[16]

J-P. Françoise and P. Yang, Quadratic double centers and their perturbations, J. Differ. Equ., 271 (2021), 563-593.  doi: 10.1016/j.jde.2020.08.035.

[17]

P. GianniB. Trager and G. Zacharias, Gröbner bases and primary decomposition of polynomials, J. Symbolic Comput., 6 (1988), 146-167.  doi: 10.1016/S0747-7171(88)80040-3.

[18]

J. Giné, Z. Kadyrsizova, Y. R. Liu and V. G. Romanovski, Linearizability conditions for Lotka-Volterra planar complex quartic systems having homogeneous nonlinearities, Comput. Math. Appl., 61 (2011), 1190–1201. doi: 10.1016/j.camwa.2010.12.069.

[19]

J. GinéJ. Llibre and C. Valls, Simultaneity of centres in $\mathbb{Z}_q$-equivariant systems, Proc. R. Soc. A., 474 (2018), 20170811.  doi: 10.1098/rspa.2017.0811.

[20]

J. Giné and C. Valls, Simultaneity of centres in double-reversible planar differential systems, Dyn. Syst., 36 (2020), 167-180.  doi: 10.1080/14689367.2020.1853061.

[21]

M. HuT. Li and X. Chen, Bi-center problem and Hopf cyclicity of a cubic Liénard system, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 401-414.  doi: 10.3934/dcdsb.2019187.

[22]

E. F. Kirnitskaya and K. S. Sibirskii, Conditions for two centers for a quadratic differential system, Diff. Uravn., 14: 9 (1978), 1589–1593.

[23]

C. Li, Planar quadratic systems possessing two centers, (in Chinese), Acta Math. Sinica., 28 (1985), 644-648. 

[24]

F. LiY. LiuY. Liu and P. Yu, Bi-center problem and bifurcation of limit cycles from nilpotent singular points in Z$_2$-equivariant cubic vector fields, J. Differ. Equ., 265 (2018), 4965-4992.  doi: 10.1016/j.jde.2018.06.027.

[25]

F. LiY. LiuY. Liu and P. Yu, Complex isochronous centers and linearization transformations for cubic Z$_2$-equivariant planar systems, J. Differ. Equ., 268 (2020), 3819-3847.  doi: 10.1016/j.jde.2019.10.011.

[26]

Y. R. Liu and J. B. Li, Complete study on a bi-center problem for the $\mathbb{Z}_2$-equivariant cubic vector fields, Acta Math. Sin., (Engl. Ser.), 27 (2011), 1379-1394.  doi: 10.1007/s10114-011-8412-8.

[27]

W. S. Loud, Behaviour of the period of solutions of certain plane autonomous systems near centers, Contributions to Differential Equations, 3 (1964), 21-36. 

[28]

P. MardešićC. Rousseau and B. Toni, Linearization of isochronous centers, J. Diff. Equa., 121 (1995), 67-108.  doi: 10.1006/jdeq.1995.1122.

[29]

G. Pfister, W. Decker, H. A. Shönemann and S. Laplagne, primdec.lib. A SINGULAR 4-2-1 library for computing the prime decomposition and radical of ideals, 2021.

[30]

I. I. Pleshkan, A new method of investigating on the isochronicity of a system of differential equations, Dokl. Akad. Nauk SSSR, 182 (1968), 768-771. 

[31]

V. G. RomanovskiX. Chen and Z. Hu, Linearizability of linear systems perturbed by fifth degree homogeneous polynomials, J. Phys. A., 40 (2007), 5905-5919.  doi: 10.1088/1751-8113/40/22/010.

[32]

V. G. RomanovskiW. Fernandes and R. Oliveira, Bi-center problem for some classes of $\mathbb Z_2$-equivariant systems, J. Comput. Appl. Math., 320 (2017), 61-75.  doi: 10.1016/j.cam.2017.02.003.

[33]

V. G. Romanovski and M. Prešern, An approach to solving systems of polynomials via modular arithmetics with applications, J. Comput. Appl. Math., 236 (2011), 196-208.  doi: 10.1016/j.cam.2011.06.018.

[34]

V. G. Romanovski and D. S. Shafer, The Center and cyclicity Problems: A computational Algebra Approach, Birkhäuser Boston, Ltd., Boston, MA, 2009. doi: 10.1007/978-0-8176-4727-8.

[35]

P. S. WangM. J. T. Guy and J. H. Davenport, P-adic reconstruction of rational numbers, SIGSAM Bull., 16 (1982), 2-3.  doi: 10.1145/1089292.1089293.

show all references

References:
[1]

J. C. Artés, F. Dumortier, C. Herssens, J. Llibre and P. De Maesschalck, Computer program P4 to study phase portraits of plane polynomial differential equation, 2003, Available from: http://mat.uab.es/ artes/p4/p4.htm.

[2]

J. ChavarrigaI. A. García and J. Giné, Isochronicity into a family of time-reversible cubic vector fields, Appl. Math. Comput., 121 (2001), 129-145.  doi: 10.1016/S0096-3003(99)00267-2.

[3]

J. ChavarrigaI. A. García and J. Giné, Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomial, Bull. Sci. Math., 123 (1999), 77-96.  doi: 10.1016/S0007-4497(99)80015-3.

[4]

J. ChavarrigaI. A. García and J. Giné, Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials, J. Comput. Appl. Math., 126 (2000), 351-368.  doi: 10.1016/S0377-0427(99)00364-7.

[5]

L. ChenZ. Lu and D. Wang, A class of cubic systems with two centers or two foci, J. Math. Anal. Appl., 242 (2000), 154-163.  doi: 10.1006/jmaa.1999.6630.

[6]

T. ChenS. Li and J. Llibre, Z$_2$-equivariant linear type bi-center cubic polynomial Hamiltonian vector fields, J. Differ. Equ., 269 (2020), 832-861.  doi: 10.1016/j.jde.2019.12.020.

[7]

X. ChenW. HuangV. G. Romanovski and W. Zhang, Linearizability conditions of a time-reversible quartic-like system, J. Math. Anal. Appl., 383 (2011), 179-189.  doi: 10.1016/j.jmaa.2011.05.018.

[8]

X. ChenV. G. Romanovski and W. Zhang, Linearizability conditions of time-reversible quartic systems having homogeneous nonlinearities, Nonlinear Anal., 69 (2008), 1525-1539.  doi: 10.1016/j.na.2007.07.009.

[9]

A. CimaA. GasullV. Mañosa and F. Mañosas, Algebraic properties of the Liapunov and period constants, Rocky Mountain J. Math., 27 (1997), 471-501.  doi: 10.1216/rmjm/1181071923.

[10]

R. Conti, On a class of cubic systems with two centers, Results Math., 14 (1988), 30-37.  doi: 10.1007/BF03323214.

[11]

W. Decker, G. M. Greuel, G. Pfister and H. A.Shönemann, SINGULAR 4-2-1 –- A Computer algebra system for polynomial computations, 2021, Available from: http://www.singular.uni-kl.de.

[12]

C. Du, The problem of bicenter and isochronicity for a class of quasi symmetric planar systems, Abstr. Appl. Anal., (2014), 482450. doi: 10.1155/2014/482450.

[13]

M. DukarićW. Fernandes and R. Oliveira, Symmetric centers on planar cubic differential systems, Nonlinear Anal., 197 (2020), 111868.  doi: 10.1016/j.na.2020.111868.

[14]

W. FernandesR. Oliveira and V. G. Romanovski, Isochronicity for a $\mathbb Z_2$-equivariant quintic system, J. Math. Anal. Appl., 467 (2018), 874-892.  doi: 10.1016/j.jmaa.2018.07.053.

[15]

W. Fernandes, V. G. Romanovski, M. Sultanova and Y. Tang, Isochronicity and linearizability of a planar cubic system, J. Math. Anal. Appl., 450 (2017), 795–813. doi: 10.1016/j.jmaa.2017.01.058.

[16]

J-P. Françoise and P. Yang, Quadratic double centers and their perturbations, J. Differ. Equ., 271 (2021), 563-593.  doi: 10.1016/j.jde.2020.08.035.

[17]

P. GianniB. Trager and G. Zacharias, Gröbner bases and primary decomposition of polynomials, J. Symbolic Comput., 6 (1988), 146-167.  doi: 10.1016/S0747-7171(88)80040-3.

[18]

J. Giné, Z. Kadyrsizova, Y. R. Liu and V. G. Romanovski, Linearizability conditions for Lotka-Volterra planar complex quartic systems having homogeneous nonlinearities, Comput. Math. Appl., 61 (2011), 1190–1201. doi: 10.1016/j.camwa.2010.12.069.

[19]

J. GinéJ. Llibre and C. Valls, Simultaneity of centres in $\mathbb{Z}_q$-equivariant systems, Proc. R. Soc. A., 474 (2018), 20170811.  doi: 10.1098/rspa.2017.0811.

[20]

J. Giné and C. Valls, Simultaneity of centres in double-reversible planar differential systems, Dyn. Syst., 36 (2020), 167-180.  doi: 10.1080/14689367.2020.1853061.

[21]

M. HuT. Li and X. Chen, Bi-center problem and Hopf cyclicity of a cubic Liénard system, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 401-414.  doi: 10.3934/dcdsb.2019187.

[22]

E. F. Kirnitskaya and K. S. Sibirskii, Conditions for two centers for a quadratic differential system, Diff. Uravn., 14: 9 (1978), 1589–1593.

[23]

C. Li, Planar quadratic systems possessing two centers, (in Chinese), Acta Math. Sinica., 28 (1985), 644-648. 

[24]

F. LiY. LiuY. Liu and P. Yu, Bi-center problem and bifurcation of limit cycles from nilpotent singular points in Z$_2$-equivariant cubic vector fields, J. Differ. Equ., 265 (2018), 4965-4992.  doi: 10.1016/j.jde.2018.06.027.

[25]

F. LiY. LiuY. Liu and P. Yu, Complex isochronous centers and linearization transformations for cubic Z$_2$-equivariant planar systems, J. Differ. Equ., 268 (2020), 3819-3847.  doi: 10.1016/j.jde.2019.10.011.

[26]

Y. R. Liu and J. B. Li, Complete study on a bi-center problem for the $\mathbb{Z}_2$-equivariant cubic vector fields, Acta Math. Sin., (Engl. Ser.), 27 (2011), 1379-1394.  doi: 10.1007/s10114-011-8412-8.

[27]

W. S. Loud, Behaviour of the period of solutions of certain plane autonomous systems near centers, Contributions to Differential Equations, 3 (1964), 21-36. 

[28]

P. MardešićC. Rousseau and B. Toni, Linearization of isochronous centers, J. Diff. Equa., 121 (1995), 67-108.  doi: 10.1006/jdeq.1995.1122.

[29]

G. Pfister, W. Decker, H. A. Shönemann and S. Laplagne, primdec.lib. A SINGULAR 4-2-1 library for computing the prime decomposition and radical of ideals, 2021.

[30]

I. I. Pleshkan, A new method of investigating on the isochronicity of a system of differential equations, Dokl. Akad. Nauk SSSR, 182 (1968), 768-771. 

[31]

V. G. RomanovskiX. Chen and Z. Hu, Linearizability of linear systems perturbed by fifth degree homogeneous polynomials, J. Phys. A., 40 (2007), 5905-5919.  doi: 10.1088/1751-8113/40/22/010.

[32]

V. G. RomanovskiW. Fernandes and R. Oliveira, Bi-center problem for some classes of $\mathbb Z_2$-equivariant systems, J. Comput. Appl. Math., 320 (2017), 61-75.  doi: 10.1016/j.cam.2017.02.003.

[33]

V. G. Romanovski and M. Prešern, An approach to solving systems of polynomials via modular arithmetics with applications, J. Comput. Appl. Math., 236 (2011), 196-208.  doi: 10.1016/j.cam.2011.06.018.

[34]

V. G. Romanovski and D. S. Shafer, The Center and cyclicity Problems: A computational Algebra Approach, Birkhäuser Boston, Ltd., Boston, MA, 2009. doi: 10.1007/978-0-8176-4727-8.

[35]

P. S. WangM. J. T. Guy and J. H. Davenport, P-adic reconstruction of rational numbers, SIGSAM Bull., 16 (1982), 2-3.  doi: 10.1145/1089292.1089293.

Figure 1.  Global phase portraits of systems (14), (16), (18), (19), (20), (21) and (23), respectively
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