doi: 10.3934/dcdsb.2021215
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.

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 Early access September 2021

Fund Project: The authors thank the reviewer for careful reading and valuable suggestions which helped to improve the manuscript.

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.

Citation: Wilker Fernandes, Viviane Pardini Valério, Patricia Tempesta. Isochronicity of bi-centers for symmetric quartic differential systems. Discrete & Continuous Dynamical Systems - B, 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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[28]

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

[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. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[28]

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

[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. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029

[2]

Jaume Llibre, Claudia Valls. Analytic integrability of a class of planar polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2657-2661. doi: 10.3934/dcdsb.2015.20.2657

[3]

Hebai Chen, Jaume Llibre, Yilei Tang. Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6495-6509. doi: 10.3934/dcdsb.2019150

[4]

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

[5]

Armengol Gasull, Hector Giacomini. Upper bounds for the number of limit cycles of some planar polynomial differential systems. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 217-229. doi: 10.3934/dcds.2010.27.217

[6]

Jaume Llibre, Roland Rabanal. Center conditions for a class of planar rigid polynomial differential systems. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1075-1090. doi: 10.3934/dcds.2015.35.1075

[7]

Yilei Tang, Long Wang, Xiang Zhang. Center of planar quintic quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2177-2191. doi: 10.3934/dcds.2015.35.2177

[8]

B. Coll, Chengzhi Li, Rafel Prohens. Quadratic perturbations of a class of quadratic reversible systems with two centers. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 699-729. doi: 10.3934/dcds.2009.24.699

[9]

Jaume Giné, Maite Grau, Jaume Llibre. Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4531-4547. doi: 10.3934/dcds.2013.33.4531

[10]

Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214

[11]

Xingwu Chen, Jaume Llibre, Weinian Zhang. Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3953-3965. doi: 10.3934/dcdsb.2017203

[12]

Montserrat Corbera, Claudia Valls. Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3209-3233. doi: 10.3934/dcdsb.2020225

[13]

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

[14]

Paul H. Rabinowitz. On a class of reversible elliptic systems. Networks & Heterogeneous Media, 2012, 7 (4) : 927-939. doi: 10.3934/nhm.2012.7.927

[15]

Yanqin Xiong, Maoan Han. Planar quasi-homogeneous polynomial systems with a given weight degree. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 4015-4025. doi: 10.3934/dcds.2016.36.4015

[16]

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

[17]

Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 887-912. doi: 10.3934/dcdsb.2018047

[18]

Shimin Li, Jaume Llibre. On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5885-5901. doi: 10.3934/dcdsb.2019111

[19]

Weigu Li, Jaume Llibre, Hao Wu. Polynomial and linearized normal forms for almost periodic differential systems. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 345-360. doi: 10.3934/dcds.2016.36.345

[20]

Tao Li, Jaume Llibre. Limit cycles of piecewise polynomial differential systems with the discontinuity line xy = 0. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3887-3909. doi: 10.3934/cpaa.2021136

2020 Impact Factor: 1.327

Article outline

Figures and Tables

[Back to Top]