Center problem for systems with two monomial nonlinearities

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March 2016, 15(2): 577-598. doi: 10.3934/cpaa.2016.15.577

Center problem for systems with two monomial nonlinearities

Armengol Gasull ^1, , Jaume Giné ^2, and Joan Torregrosa ^3,

1.	Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193-Bellaterra
2.	Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida
3.	Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona

Received August 2015 Revised December 2015 Published January 2016

Abstract
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We study the center problem for planar systems with a linear center at the origin that in complex coordinates have a nonlinearity formed by the sum of two monomials. Our first result lists several centers inside this family. To the best of our knowledge this list includes a new class of Darboux centers that are also persistent centers. The rest of the paper is dedicated to try to prove that the given list is exhaustive. We get several partial results that seem to indicate that this is the case. In particular, we solve the question for several general families with arbitrary high degree and for all cases of degree less or equal than 19. As a byproduct of our study we also obtain the highest known order for weak-foci of planar polynomial systems of some given degrees.

Keywords: Poincaré--Lyapunov constants, Nondegenerate center, Darboux center, Holomorphic center, Reversible center, Persistent center..

Mathematics Subject Classification: Primary: 34C25; Secondary: 34C14, 37C10, 37C2.

Citation: Armengol Gasull, Jaume Giné, Joan Torregrosa. Center problem for systems with two monomial nonlinearities. Communications on Pure and Applied Analysis, 2016, 15 (2) : 577-598. doi: 10.3934/cpaa.2016.15.577

References:

[1]	A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maĭer, Theory of Bifurcations of Dynamic Systems on a Plane, Halsted Press [A division of John Wiley & Sons], New York-Toronto, Ont.; Israel Program for Scientific Translations, Jerusalem-London, 1973, Translated from the Russian.
[2]	J. Bai and Y. Liu, A class of planar degree $n$ (even number) polynomial systems with a fine focus of order $n^2-n$, Chinese Sci. Bull., 12 (1992), 1063-1065, in Chinese.
[3]	A. Cima, A. Gasull, V. 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.
[4]	A. Cima, A. Gasull and J. C. Medrado, On persistent centers, Bull. Sci. Math., 133 (2009), 644-657. doi: 10.1016/j.bulsci.2008.08.007.
[5]	J. Devlin, Word problems related to derivatives of the displacement map, Math. Proc. Cambridge Philos. Soc., 110 (1991), 569-579. doi: 10.1017/S0305004100070638.
[6]	F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006.
[7]	J.-P. Françoise, Successive derivatives of a first return map, application to the study of quadratic vector fields, Ergodic Theory Dynam. Systems, 16 (1996), 87-96. doi: 10.1017/S0143385700008725.
[8]	A. Garijo, A. Gasull and X. Jarque, Normal forms for singularities of one dimensional holomorphic vector fields,, \emph{Electron. J. Differential Equations}, 2004 ().
[9]	A. Gasull, A. Guillamon and V. Mañosa, An explicit expression of the first Liapunov and period constants with applications, J. Math. Anal. Appl., 211 (1997), 190-212. doi: 10.1006/jmaa.1997.5455.
[10]	A. Gasull and J. Torregrosa, A new approach to the computation of the Lyapunov constants, Comput. Appl. Math., 20 (2001), 149-177, the geometry of differential equations and dynamical systems.
[11]	J. Giné, The center problem for a linear center perturbed by homogeneous polynomials, Acta Math. Sin. (Engl. Ser.), 22 (2006), 1613-1620. doi: 10.1007/s10114-005-0623-4.
[12]	J. Giné and X. Santallusia, On the Poincaré-Lyapunov constants and the Poincaré series, Appl. Math. (Warsaw), 28 (2001), 17-30. doi: 10.4064/am28-1-2.
[13]	J. Giné and X. Santallusia, Implementation of a new algorithm of computation of the Poincaré-Liapunov constants, J. Comput. Appl. Math., 166 (2004), 465-476. doi: 10.1016/j.cam.2003.08.043.
[14]	Y. R. Liu and J. B. Li, Theory of values of singular point in complex autonomous differential systems, Sci. China Ser. A, 33 (1990), 10-23.
[15]	J. Llibre, Integrability of polynomial differential systems, in Handbook of Differential Equations (Ordinary Differential Equations Volume I), Elsevier, Northholland, 2004, 437-532.
[16]	J. Llibre and R. Rabanal, Planar real polynomial differential systems of degree $n>3$ having a weak focus of high order, Rocky Mountain J. Math., 42 (2012), 657-693. doi: 10.1216/RMJ-2012-42-2-657.
[17]	J. Llibre and C. Valls, Centers for polynomial vector fields of arbitrary degree, Commun. Pure Appl. Anal., 8 (2009), 725-742. doi: 10.3934/cpaa.2009.8.725.
[18]	N. G. Lloyd, J. M. Pearson and V. A. Romanovsky, Computing integrability conditions for a cubic differential system, Comput. Math. Appl., 32 (1996), 99-107. doi: 10.1016/S0898-1221(96)00188-5.
[19]	A. M. Lyapunov, The general problem of the stability of motion, Taylor & Francis, Ltd., London, 1992, Translated from Edouard Davaux's French translation (1907) of the 1892 Russian original and edited by A. T. Fuller. Reprint of Internat. J. Control 5 (1992). doi: 10.1080/00207179208934253.
[20]	J. M. Pearson, N. G. Lloyd and C. J. Christopher, Algorithmic derivation of centre conditions, SIAM Rev., 38 (1996), 619-636. doi: 10.1137/S0036144595283575.
[21]	H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré I, Rend. Circ. Mat. Palermo, 5 (1891), 161-191.
[22]	H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré {II}, Rend. Circ. Mat. Palermo, 11 (1897), 193-239.
[23]	Y. Qiu and J. Yang, On the focus order of planar polynomial differential equations, J. Differential Equations, 246 (2009), 3361-3379. doi: 10.1016/j.jde.2009.02.005.
[24]	V. G. Romanovskiĭ, Center conditions for a cubic system with four complex parameters, Differentsial\cprime nye Uravneniya, 31 (1995), 1091-1093, 1104; translation in Differential Equations, 31 (1995), 1023-1026.
[25]	V. G. Romanovskiĭ and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4727-8.
[26]	S. L. Shi, A method of constructing cycles without contact around a weak focus, J. Differential Equations, 41 (1981), 301-312. doi: 10.1016/0022-0396(81)90039-5.
[27]	H. Żoładek, Quadratic systems with center and their perturbations, J. Differential Equations, 109 (1994), 223-273. doi: 10.1006/jdeq.1994.1049.

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References:

[1]	A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maĭer, Theory of Bifurcations of Dynamic Systems on a Plane, Halsted Press [A division of John Wiley & Sons], New York-Toronto, Ont.; Israel Program for Scientific Translations, Jerusalem-London, 1973, Translated from the Russian.
[2]	J. Bai and Y. Liu, A class of planar degree $n$ (even number) polynomial systems with a fine focus of order $n^2-n$, Chinese Sci. Bull., 12 (1992), 1063-1065, in Chinese.
[3]	A. Cima, A. Gasull, V. 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.
[4]	A. Cima, A. Gasull and J. C. Medrado, On persistent centers, Bull. Sci. Math., 133 (2009), 644-657. doi: 10.1016/j.bulsci.2008.08.007.
[5]	J. Devlin, Word problems related to derivatives of the displacement map, Math. Proc. Cambridge Philos. Soc., 110 (1991), 569-579. doi: 10.1017/S0305004100070638.
[6]	F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006.
[7]	J.-P. Françoise, Successive derivatives of a first return map, application to the study of quadratic vector fields, Ergodic Theory Dynam. Systems, 16 (1996), 87-96. doi: 10.1017/S0143385700008725.
[8]	A. Garijo, A. Gasull and X. Jarque, Normal forms for singularities of one dimensional holomorphic vector fields,, \emph{Electron. J. Differential Equations}, 2004 ().
[9]	A. Gasull, A. Guillamon and V. Mañosa, An explicit expression of the first Liapunov and period constants with applications, J. Math. Anal. Appl., 211 (1997), 190-212. doi: 10.1006/jmaa.1997.5455.
[10]	A. Gasull and J. Torregrosa, A new approach to the computation of the Lyapunov constants, Comput. Appl. Math., 20 (2001), 149-177, the geometry of differential equations and dynamical systems.
[11]	J. Giné, The center problem for a linear center perturbed by homogeneous polynomials, Acta Math. Sin. (Engl. Ser.), 22 (2006), 1613-1620. doi: 10.1007/s10114-005-0623-4.
[12]	J. Giné and X. Santallusia, On the Poincaré-Lyapunov constants and the Poincaré series, Appl. Math. (Warsaw), 28 (2001), 17-30. doi: 10.4064/am28-1-2.
[13]	J. Giné and X. Santallusia, Implementation of a new algorithm of computation of the Poincaré-Liapunov constants, J. Comput. Appl. Math., 166 (2004), 465-476. doi: 10.1016/j.cam.2003.08.043.
[14]	Y. R. Liu and J. B. Li, Theory of values of singular point in complex autonomous differential systems, Sci. China Ser. A, 33 (1990), 10-23.
[15]	J. Llibre, Integrability of polynomial differential systems, in Handbook of Differential Equations (Ordinary Differential Equations Volume I), Elsevier, Northholland, 2004, 437-532.
[16]	J. Llibre and R. Rabanal, Planar real polynomial differential systems of degree $n>3$ having a weak focus of high order, Rocky Mountain J. Math., 42 (2012), 657-693. doi: 10.1216/RMJ-2012-42-2-657.
[17]	J. Llibre and C. Valls, Centers for polynomial vector fields of arbitrary degree, Commun. Pure Appl. Anal., 8 (2009), 725-742. doi: 10.3934/cpaa.2009.8.725.
[18]	N. G. Lloyd, J. M. Pearson and V. A. Romanovsky, Computing integrability conditions for a cubic differential system, Comput. Math. Appl., 32 (1996), 99-107. doi: 10.1016/S0898-1221(96)00188-5.
[19]	A. M. Lyapunov, The general problem of the stability of motion, Taylor & Francis, Ltd., London, 1992, Translated from Edouard Davaux's French translation (1907) of the 1892 Russian original and edited by A. T. Fuller. Reprint of Internat. J. Control 5 (1992). doi: 10.1080/00207179208934253.
[20]	J. M. Pearson, N. G. Lloyd and C. J. Christopher, Algorithmic derivation of centre conditions, SIAM Rev., 38 (1996), 619-636. doi: 10.1137/S0036144595283575.
[21]	H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré I, Rend. Circ. Mat. Palermo, 5 (1891), 161-191.
[22]	H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré {II}, Rend. Circ. Mat. Palermo, 11 (1897), 193-239.
[23]	Y. Qiu and J. Yang, On the focus order of planar polynomial differential equations, J. Differential Equations, 246 (2009), 3361-3379. doi: 10.1016/j.jde.2009.02.005.
[24]	V. G. Romanovskiĭ, Center conditions for a cubic system with four complex parameters, Differentsial\cprime nye Uravneniya, 31 (1995), 1091-1093, 1104; translation in Differential Equations, 31 (1995), 1023-1026.
[25]	V. G. Romanovskiĭ and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4727-8.
[26]	S. L. Shi, A method of constructing cycles without contact around a weak focus, J. Differential Equations, 41 (1981), 301-312. doi: 10.1016/0022-0396(81)90039-5.
[27]	H. Żoładek, Quadratic systems with center and their perturbations, J. Differential Equations, 109 (1994), 223-273. doi: 10.1006/jdeq.1994.1049.

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