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Concentration of solutions for the fractional Nirenberg problem
Center problem for systems with two monomial nonlinearities
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 |
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. |
show all references
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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