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

Center conditions for generalized polynomial kukles systems

Jaume Giné

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.

Keywords: Center problem, analytic integrability, polynomial Generalized Kukles systems, Gröbner bases, decomposition in prime ideals.
Mathematics Subject Classification: Primary: 34C05; Secondary: 37C10, 34C25, 34C07.
Citation: Jaume Giné. Center conditions for generalized polynomial kukles systems. Communications on Pure & 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.   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.   Google Scholar

[3]

L. A. Cherkas, Conditions for a center for a certain Lienard equation, Differ. Uravn., 12 (1976), 292-298.   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.   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, Giné J., 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]

Giné J., 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]

Giné J. and 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.   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.   Google Scholar

[3]

L. A. Cherkas, Conditions for a center for a certain Lienard equation, Differ. Uravn., 12 (1976), 292-298.   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.   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, Giné J., 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]

Giné J., 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]

Giné J. and 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

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