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Long-term stability for kdv solitons in weighted Hs spaces
Center conditions for generalized polynomial kukles systems
Departament de Matemàtica, Universitat de Lleida, Avda. Jaume Ⅱ, 69; 25001 Lleida, Catalonia, Spain |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
C. J. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser-Verlag, Basel, 2007. |
[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. |
[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. |
[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. |
[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. |
[11] |
J. Giné, Singularity analysis in planar vector fields, J. Math. Phys., 55 (2014), 112703.
doi: 10.1063/1.4901544. |
[12] |
J. Giné, Center conditions for polynomial Liénard systems, Qual. Theory Dyn. Syst. , to appear.
doi: 10.1007/s12346-016-0202-3. |
[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. |
[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. |
[15] |
I. S. Kukles, Sur quelques cas de distinction entre un foyer et un centre, Dolk. Akad. Nauk SSSR, 42 (1944), 208-211. |
[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). |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
P. S. Wang, M. J. T. Guy and J. H. Davenport, P-adic reconstruction of rational numbers, SIGSAM Bull., 16 (1982), 2-3. |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
C. J. Christopher and C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser-Verlag, Basel, 2007. |
[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. |
[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. |
[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. |
[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. |
[11] |
J. Giné, Singularity analysis in planar vector fields, J. Math. Phys., 55 (2014), 112703.
doi: 10.1063/1.4901544. |
[12] |
J. Giné, Center conditions for polynomial Liénard systems, Qual. Theory Dyn. Syst. , to appear.
doi: 10.1007/s12346-016-0202-3. |
[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. |
[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. |
[15] |
I. S. Kukles, Sur quelques cas de distinction entre un foyer et un centre, Dolk. Akad. Nauk SSSR, 42 (1944), 208-211. |
[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). |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[26] |
P. S. Wang, M. J. T. Guy and J. H. Davenport, P-adic reconstruction of rational numbers, SIGSAM Bull., 16 (1982), 2-3. |
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