# American Institute of Mathematical Sciences

March  2017, 16(2): 417-426. doi: 10.3934/cpaa.2017021

## Center conditions for generalized polynomial kukles systems

 Departament de Matemàtica, Universitat de Lleida, Avda. Jaume Ⅱ, 69; 25001 Lleida, Catalonia, Spain

Received  January 2016 Revised  October 2016 Published  January 2017

Fund Project: The author is partially supported by a MINECO/FEDER grant number MTM2014-53703-P and by a AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204.

Abstract. In this paper we study the center problem for certain generalized Kukles systems $\dot{x}= y, \qquad \dot{y}= P_0(x)+ P_1(x)y+P_2(x) y^2+ P_3(x) y^3,$\end{document} where Pi(x) are polynomials of degree n, P0(0) = 0 and P0′(0) < 0. Computing the focal values and using modular arithmetics and Gröbner bases we find the center conditions for such systems when P0 is of degree 2 and Pi for i = 1; 2; 3 are of degree 3 without constant terms. We also establish a conjecture about the center conditions for such systems.

Citation: Jaume Giné. Center conditions for generalized polynomial kukles systems. Communications on Pure &amp; Applied Analysis, 2017, 16 (2) : 417-426. doi: 10.3934/cpaa.2017021
##### References:
 [1] L. A. Cherkas, On the conditions for a center for certain equations of the form yy′ = P(x) + Q(x)y + R(x)y2, Differ. Uravn., 8 (1972), 1435-1439; Differ. Equ., 8 (1972), 1104-1107.  Google Scholar [2] L. A. Cherkas, Conditions for a center for the equation $P_3(x) yy'=\sum_{i=0}^2 P_i(x)y^i$ , Differ. Uravn., 10 (1974), 367-368; Differ. Equ., 10 (1974), 276-277.  Google Scholar [3] L. A. Cherkas, Conditions for a center for a certain Lienard equation, Differ. Uravn., 12 (1976), 292-298; Differ. Equ., 12 (1976), 201-206.  Google Scholar [4] L. A. Cherkas, Conditions for the equation $yy'=\sum_{i=0}^3 P_i(x)y^i$ to have a center, Differ. Uravn., 14 (1978), 1594-1600; Differ. Equ., 14 (1978), 1133-1137.  Google Scholar [5] C. J. Christopher, An algebraic approach to the classification of centres in polynomial Liénard systems, J. Math. Anal. Appl., 229 (1999), 319-329. doi: 10.1006/jmaa.1998.6175.  Google Scholar [6] C. J. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser-Verlag, Basel, 2007.  Google Scholar [7] C. J. 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 [8] W. Decker, S. Laplagne, G. Pfister and H. A. Schonemann, SINGULAR, 3-1 library for computing the prime decomposition and radical of ideals, primdec. lib, 2010. Google Scholar [9] B. FerČec, J. Giné, V. G. Romanovski and V. F. Edneral, Integrability of complex planar systems with homogeneous nonlinearities, J. Math. Anal. Appl., 434 (2016), 894-914. doi: 10.1016/j.jmaa.2015.09.037.  Google Scholar [10] P. Gianni, B. Trager and G. Zacharias, Gröbner bases and primary decompositions of polynomials, J. Symbolic Comput., 6 (1988) 146-167. doi: 10.1016/S0747-7171(88)80040-3.  Google Scholar [11] J. Giné, Singularity analysis in planar vector fields, J. Math. Phys., 55 (2014), 112703. doi: 10.1063/1.4901544.  Google Scholar [12] J. Giné, Center conditions for polynomial Liénard systems, Qual. Theory Dyn. Syst. , to appear. doi: 10.1007/s12346-016-0202-3.  Google Scholar [13] J. Giné, J. Llibre, Analytic reducibility of nondegenerate centers: Cherkas systems, Electron. J. Qual. Theory Differ. Equ., 49 (2016), 1-10. doi: 10.14232/ejqtde.2016.1.49.  Google Scholar [14] G. M. Greuel, G. Pfister and H. A. Schönemann, SINGULAR 3. 0, A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserlautern (2005). http://www.singular.uni-kl.de. Google Scholar [15] I. S. Kukles, Sur quelques cas de distinction entre un foyer et un centre, Dolk. Akad. Nauk SSSR, 42 (1944), 208-211.  Google Scholar [16] A. A. Kushner and A. P. Sadovskii, Center conditions for Lienard-type systems of degree four, Vestn. Beloruss. Gos. Univ. Ser. 1 Fiz. Mat. Inform. , (2011), 119-122 (Russian).  Google Scholar [17] J. Llibre and R. Rabanal, Center conditions for a class of planar rigid polynomial differential systems, Discrete Contin. Dyn. Syst., 35 (2015), 1075-1090. doi: 10.3934/dcds.2015.35.1075.  Google Scholar [18] N. G. Lloyd and J. M. Pearson, Computing centre conditions for certain cubic systems, J. Comp. Appl. Math., 40 (1992), 323-336. doi: 10.1016/0377-0427(92)90188-4.  Google Scholar [19] J. M. Pearson and N. G. Lloyd, Kukles revisited: Advances in computing techniques, Comp. Math. Appl., 60 (2010), 2797-2805. doi: 10.1016/j.camwa.2010.09.034.  Google Scholar [20] 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 [21] V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhauser Boston, Inc. , Boston, MA, 2009. doi: 10.1007/978-0-8176-4727-8.  Google Scholar [22] A. P. Sadovskii, Solution of the center and focus problem for a cubic system of nonlinear oscillations, Differ. Uravn. , 33 (1997), 236-244 (Russian); Differential Equations, 33 (1997), 236-244.  Google Scholar [23] A. P. Sadovskii, On conditions for a center and focus for nonlinear oscillation equations, Differ. Uravn. , 15 (1979), 1716-1719 (Russian); Differential Equations, 15 (1979), 1226-1229.  Google Scholar [24] A. P. Sadovskii and T. V. Shcheglova, Solution of the center-focus problem for a cubic system with nine parameters, Differ. Uravn. , 47 (2011), 209-224 (Russian); Differential Equations, 47 (2011), 208-223. doi: 10.1134/S0012266111020078.  Google Scholar [25] A. P. Sadovskii and T. V. Shcheglova, Center conditions for a polynomial differential system, Differ. Uravn. , 49 (2013), 151-164; Differencial Equations, 49 (2013), 151-165. doi: 10.1134/S001226611302002X.  Google Scholar [26] P. S. Wang, M. J. T. Guy and J. H. Davenport, P-adic reconstruction of rational numbers, SIGSAM Bull., 16 (1982), 2-3. Google Scholar

show all references

##### References:
 [1] L. A. Cherkas, On the conditions for a center for certain equations of the form yy′ = P(x) + Q(x)y + R(x)y2, Differ. Uravn., 8 (1972), 1435-1439; Differ. Equ., 8 (1972), 1104-1107.  Google Scholar [2] L. A. Cherkas, Conditions for a center for the equation $P_3(x) yy'=\sum_{i=0}^2 P_i(x)y^i$ , Differ. Uravn., 10 (1974), 367-368; Differ. Equ., 10 (1974), 276-277.  Google Scholar [3] L. A. Cherkas, Conditions for a center for a certain Lienard equation, Differ. Uravn., 12 (1976), 292-298; Differ. Equ., 12 (1976), 201-206.  Google Scholar [4] L. A. Cherkas, Conditions for the equation $yy'=\sum_{i=0}^3 P_i(x)y^i$ to have a center, Differ. Uravn., 14 (1978), 1594-1600; Differ. Equ., 14 (1978), 1133-1137.  Google Scholar [5] C. J. Christopher, An algebraic approach to the classification of centres in polynomial Liénard systems, J. Math. Anal. Appl., 229 (1999), 319-329. doi: 10.1006/jmaa.1998.6175.  Google Scholar [6] C. J. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser-Verlag, Basel, 2007.  Google Scholar [7] C. J. 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 [8] W. Decker, S. Laplagne, G. Pfister and H. A. Schonemann, SINGULAR, 3-1 library for computing the prime decomposition and radical of ideals, primdec. lib, 2010. Google Scholar [9] B. FerČec, J. Giné, V. G. Romanovski and V. F. Edneral, Integrability of complex planar systems with homogeneous nonlinearities, J. Math. Anal. Appl., 434 (2016), 894-914. doi: 10.1016/j.jmaa.2015.09.037.  Google Scholar [10] P. Gianni, B. Trager and G. Zacharias, Gröbner bases and primary decompositions of polynomials, J. Symbolic Comput., 6 (1988) 146-167. doi: 10.1016/S0747-7171(88)80040-3.  Google Scholar [11] J. Giné, Singularity analysis in planar vector fields, J. Math. Phys., 55 (2014), 112703. doi: 10.1063/1.4901544.  Google Scholar [12] J. Giné, Center conditions for polynomial Liénard systems, Qual. Theory Dyn. Syst. , to appear. doi: 10.1007/s12346-016-0202-3.  Google Scholar [13] J. Giné, J. Llibre, Analytic reducibility of nondegenerate centers: Cherkas systems, Electron. J. Qual. Theory Differ. Equ., 49 (2016), 1-10. doi: 10.14232/ejqtde.2016.1.49.  Google Scholar [14] G. M. Greuel, G. Pfister and H. A. Schönemann, SINGULAR 3. 0, A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserlautern (2005). http://www.singular.uni-kl.de. Google Scholar [15] I. S. Kukles, Sur quelques cas de distinction entre un foyer et un centre, Dolk. Akad. Nauk SSSR, 42 (1944), 208-211.  Google Scholar [16] A. A. Kushner and A. P. Sadovskii, Center conditions for Lienard-type systems of degree four, Vestn. Beloruss. Gos. Univ. Ser. 1 Fiz. Mat. Inform. , (2011), 119-122 (Russian).  Google Scholar [17] J. Llibre and R. Rabanal, Center conditions for a class of planar rigid polynomial differential systems, Discrete Contin. Dyn. Syst., 35 (2015), 1075-1090. doi: 10.3934/dcds.2015.35.1075.  Google Scholar [18] N. G. Lloyd and J. M. Pearson, Computing centre conditions for certain cubic systems, J. Comp. Appl. Math., 40 (1992), 323-336. doi: 10.1016/0377-0427(92)90188-4.  Google Scholar [19] J. M. Pearson and N. G. Lloyd, Kukles revisited: Advances in computing techniques, Comp. Math. Appl., 60 (2010), 2797-2805. doi: 10.1016/j.camwa.2010.09.034.  Google Scholar [20] 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 [21] V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhauser Boston, Inc. , Boston, MA, 2009. doi: 10.1007/978-0-8176-4727-8.  Google Scholar [22] A. P. Sadovskii, Solution of the center and focus problem for a cubic system of nonlinear oscillations, Differ. Uravn. , 33 (1997), 236-244 (Russian); Differential Equations, 33 (1997), 236-244.  Google Scholar [23] A. P. Sadovskii, On conditions for a center and focus for nonlinear oscillation equations, Differ. Uravn. , 15 (1979), 1716-1719 (Russian); Differential Equations, 15 (1979), 1226-1229.  Google Scholar [24] A. P. Sadovskii and T. V. Shcheglova, Solution of the center-focus problem for a cubic system with nine parameters, Differ. Uravn. , 47 (2011), 209-224 (Russian); Differential Equations, 47 (2011), 208-223. doi: 10.1134/S0012266111020078.  Google Scholar [25] A. P. Sadovskii and T. V. Shcheglova, Center conditions for a polynomial differential system, Differ. Uravn. , 49 (2013), 151-164; Differencial Equations, 49 (2013), 151-165. doi: 10.1134/S001226611302002X.  Google Scholar [26] P. S. Wang, M. J. T. Guy and J. H. Davenport, P-adic reconstruction of rational numbers, SIGSAM Bull., 16 (1982), 2-3. Google Scholar
 [1] Ismara Álvarez-Barrientos, Mijail Borges-Quintana, Miguel Angel Borges-Trenard, Daniel Panario. Computing Gröbner bases associated with lattices. Advances in Mathematics of Communications, 2016, 10 (4) : 851-860. doi: 10.3934/amc.2016045 [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] Hannes Bartz, Antonia Wachter-Zeh. Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases. Advances in Mathematics of Communications, 2018, 12 (4) : 773-804. doi: 10.3934/amc.2018046 [4] Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 557-573. doi: 10.3934/dcdsb.2016.21.557 [5] Arnulf Jentzen, Felix Lindner, Primož Pušnik. On the Alekseev-Gröbner formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4475-4511. doi: 10.3934/dcdsb.2019128 [6] Antonio Algaba, Cristóbal García, Jaume Giné. Analytic integrability for some degenerate planar systems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2797-2809. doi: 10.3934/cpaa.2013.12.2797 [7] 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 [8] 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 [9] Armengol Gasull, Jaume Giné, Joan Torregrosa. Center problem for systems with two monomial nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (2) : 577-598. doi: 10.3934/cpaa.2016.15.577 [10] G.F. Webb. The prime number periodical cicada problem. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 387-399. doi: 10.3934/dcdsb.2001.1.387 [11] Salomón Rebollo-Perdomo, Claudio Vidal. Bifurcation of limit cycles for a family of perturbed Kukles differential systems. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4189-4202. doi: 10.3934/dcds.2018182 [12] Fabio Scalco Dias, Luis Fernando Mello. The center--focus problem and small amplitude limit cycles in rigid systems. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1627-1637. doi: 10.3934/dcds.2012.32.1627 [13] Balázs Boros, Josef Hofbauer, Stefan Müller, Georg Regensburger. Planar S-systems: Global stability and the center problem. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 707-727. doi: 10.3934/dcds.2019029 [14] Isaac A. García, Douglas S. Shafer. Cyclicity of a class of polynomial nilpotent center singularities. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2497-2520. doi: 10.3934/dcds.2016.36.2497 [15] Antonio Algaba, María Díaz, Cristóbal García, Jaume Giné. Analytic integrability around a nilpotent singularity: The non-generic case. Communications on Pure & Applied Analysis, 2020, 19 (1) : 407-423. doi: 10.3934/cpaa.2020021 [16] Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085 [17] Primitivo B. Acosta-Humánez, J. Tomás Lázaro, Juan J. Morales-Ruiz, Chara Pantazi. On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1767-1800. doi: 10.3934/dcds.2015.35.1767 [18] Jaume Llibre, Yilei Tang. Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1769-1784. doi: 10.3934/dcdsb.2018236 [19] Vladimir S. Gerdjikov, Rossen I. Ivanov, Aleksander A. Stefanov. Riemann-Hilbert problem, integrability and reductions. Journal of Geometric Mechanics, 2019, 11 (2) : 167-185. doi: 10.3934/jgm.2019009 [20] Tongjiang Yan, Yanyan Liu, Yuhua Sun. Cyclic codes from two-prime generalized cyclotomic sequences of order 6. Advances in Mathematics of Communications, 2016, 10 (4) : 707-723. doi: 10.3934/amc.2016036

2020 Impact Factor: 1.916