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Existence and concentration of semiclassical solutions for Hamiltonian elliptic system
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], (1973).
|
[2] |
J. Bai and Y. Liu, A class of planar degree $n$ (even number) polynomial systems with a fine focus of order $n^2-n$,, \emph{Chinese Sci. Bull.}, 12 (1992), 1063. |
[3] |
A. Cima, A. Gasull, V. Mañosa and F. Mañosas, Algebraic properties of the Liapunov and period constants,, \emph{Rocky Mountain J. Math.}, 27 (1997), 471.
doi: 10.1216/rmjm/1181071923. |
[4] |
A. Cima, A. Gasull and J. C. Medrado, On persistent centers,, \emph{Bull. Sci. Math.}, 133 (2009), 644.
doi: 10.1016/j.bulsci.2008.08.007. |
[5] |
J. Devlin, Word problems related to derivatives of the displacement map,, \emph{Math. Proc. Cambridge Philos. Soc.}, 110 (1991), 569.
doi: 10.1017/S0305004100070638. |
[6] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems,, Universitext, (2006).
|
[7] |
J.-P. Françoise, Successive derivatives of a first return map, application to the study of quadratic vector fields,, \emph{Ergodic Theory Dynam. Systems}, 16 (1996), 87.
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,, \emph{J. Math. Anal. Appl.}, 211 (1997), 190.
doi: 10.1006/jmaa.1997.5455. |
[10] |
A. Gasull and J. Torregrosa, A new approach to the computation of the Lyapunov constants,, \emph{Comput. Appl. Math.}, 20 (2001), 149.
|
[11] |
J. Giné, The center problem for a linear center perturbed by homogeneous polynomials,, \emph{Acta Math. Sin. (Engl. Ser.)}, 22 (2006), 1613.
doi: 10.1007/s10114-005-0623-4. |
[12] |
J. Giné and X. Santallusia, On the Poincaré-Lyapunov constants and the Poincaré series,, \emph{Appl. Math. (Warsaw)}, 28 (2001), 17.
doi: 10.4064/am28-1-2. |
[13] |
J. Giné and X. Santallusia, Implementation of a new algorithm of computation of the Poincaré-Liapunov constants,, \emph{J. Comput. Appl. Math.}, 166 (2004), 465.
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,, \emph{Sci. China Ser. A}, 33 (1990), 10.
|
[15] |
J. Llibre, Integrability of polynomial differential systems,, in \emph{Handbook of Differential Equations (Ordinary Differential Equations Volume I)}, (2004), 437.
|
[16] |
J. Llibre and R. Rabanal, Planar real polynomial differential systems of degree $n>3$ having a weak focus of high order,, \emph{Rocky Mountain J. Math.}, 42 (2012), 657.
doi: 10.1216/RMJ-2012-42-2-657. |
[17] |
J. Llibre and C. Valls, Centers for polynomial vector fields of arbitrary degree,, \emph{Commun. Pure Appl. Anal.}, 8 (2009), 725.
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,, \emph{Comput. Math. Appl.}, 32 (1996), 99.
doi: 10.1016/S0898-1221(96)00188-5. |
[19] |
A. M. Lyapunov, The general problem of the stability of motion,, Taylor & Francis, 5 (1907).
doi: 10.1080/00207179208934253. |
[20] |
J. M. Pearson, N. G. Lloyd and C. J. Christopher, Algorithmic derivation of centre conditions,, \emph{SIAM Rev.}, 38 (1996), 619.
doi: 10.1137/S0036144595283575. |
[21] |
H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré I,, \emph{Rend. Circ. Mat. Palermo}, 5 (1891), 161. |
[22] |
H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré {II},, \emph{Rend. Circ. Mat. Palermo}, 11 (1897), 193. |
[23] |
Y. Qiu and J. Yang, On the focus order of planar polynomial differential equations,, \emph{J. Differential Equations}, 246 (2009), 3361.
doi: 10.1016/j.jde.2009.02.005. |
[24] |
V. G. Romanovskiĭ, Center conditions for a cubic system with four complex parameters,, \emph{Differentsial\cprime nye Uravneniya}, 31 (1995), 1091.
|
[25] |
V. G. Romanovskiĭ and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach,, Birkh\, (2009).
doi: 10.1007/978-0-8176-4727-8. |
[26] |
S. L. Shi, A method of constructing cycles without contact around a weak focus,, \emph{J. Differential Equations}, 41 (1981), 301.
doi: 10.1016/0022-0396(81)90039-5. |
[27] |
H. Żoładek, Quadratic systems with center and their perturbations,, \emph{J. Differential Equations}, 109 (1994), 223.
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], (1973).
|
[2] |
J. Bai and Y. Liu, A class of planar degree $n$ (even number) polynomial systems with a fine focus of order $n^2-n$,, \emph{Chinese Sci. Bull.}, 12 (1992), 1063. |
[3] |
A. Cima, A. Gasull, V. Mañosa and F. Mañosas, Algebraic properties of the Liapunov and period constants,, \emph{Rocky Mountain J. Math.}, 27 (1997), 471.
doi: 10.1216/rmjm/1181071923. |
[4] |
A. Cima, A. Gasull and J. C. Medrado, On persistent centers,, \emph{Bull. Sci. Math.}, 133 (2009), 644.
doi: 10.1016/j.bulsci.2008.08.007. |
[5] |
J. Devlin, Word problems related to derivatives of the displacement map,, \emph{Math. Proc. Cambridge Philos. Soc.}, 110 (1991), 569.
doi: 10.1017/S0305004100070638. |
[6] |
F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems,, Universitext, (2006).
|
[7] |
J.-P. Françoise, Successive derivatives of a first return map, application to the study of quadratic vector fields,, \emph{Ergodic Theory Dynam. Systems}, 16 (1996), 87.
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,, \emph{J. Math. Anal. Appl.}, 211 (1997), 190.
doi: 10.1006/jmaa.1997.5455. |
[10] |
A. Gasull and J. Torregrosa, A new approach to the computation of the Lyapunov constants,, \emph{Comput. Appl. Math.}, 20 (2001), 149.
|
[11] |
J. Giné, The center problem for a linear center perturbed by homogeneous polynomials,, \emph{Acta Math. Sin. (Engl. Ser.)}, 22 (2006), 1613.
doi: 10.1007/s10114-005-0623-4. |
[12] |
J. Giné and X. Santallusia, On the Poincaré-Lyapunov constants and the Poincaré series,, \emph{Appl. Math. (Warsaw)}, 28 (2001), 17.
doi: 10.4064/am28-1-2. |
[13] |
J. Giné and X. Santallusia, Implementation of a new algorithm of computation of the Poincaré-Liapunov constants,, \emph{J. Comput. Appl. Math.}, 166 (2004), 465.
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,, \emph{Sci. China Ser. A}, 33 (1990), 10.
|
[15] |
J. Llibre, Integrability of polynomial differential systems,, in \emph{Handbook of Differential Equations (Ordinary Differential Equations Volume I)}, (2004), 437.
|
[16] |
J. Llibre and R. Rabanal, Planar real polynomial differential systems of degree $n>3$ having a weak focus of high order,, \emph{Rocky Mountain J. Math.}, 42 (2012), 657.
doi: 10.1216/RMJ-2012-42-2-657. |
[17] |
J. Llibre and C. Valls, Centers for polynomial vector fields of arbitrary degree,, \emph{Commun. Pure Appl. Anal.}, 8 (2009), 725.
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,, \emph{Comput. Math. Appl.}, 32 (1996), 99.
doi: 10.1016/S0898-1221(96)00188-5. |
[19] |
A. M. Lyapunov, The general problem of the stability of motion,, Taylor & Francis, 5 (1907).
doi: 10.1080/00207179208934253. |
[20] |
J. M. Pearson, N. G. Lloyd and C. J. Christopher, Algorithmic derivation of centre conditions,, \emph{SIAM Rev.}, 38 (1996), 619.
doi: 10.1137/S0036144595283575. |
[21] |
H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré I,, \emph{Rend. Circ. Mat. Palermo}, 5 (1891), 161. |
[22] |
H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré {II},, \emph{Rend. Circ. Mat. Palermo}, 11 (1897), 193. |
[23] |
Y. Qiu and J. Yang, On the focus order of planar polynomial differential equations,, \emph{J. Differential Equations}, 246 (2009), 3361.
doi: 10.1016/j.jde.2009.02.005. |
[24] |
V. G. Romanovskiĭ, Center conditions for a cubic system with four complex parameters,, \emph{Differentsial\cprime nye Uravneniya}, 31 (1995), 1091.
|
[25] |
V. G. Romanovskiĭ and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach,, Birkh\, (2009).
doi: 10.1007/978-0-8176-4727-8. |
[26] |
S. L. Shi, A method of constructing cycles without contact around a weak focus,, \emph{J. Differential Equations}, 41 (1981), 301.
doi: 10.1016/0022-0396(81)90039-5. |
[27] |
H. Żoładek, Quadratic systems with center and their perturbations,, \emph{J. Differential Equations}, 109 (1994), 223.
doi: 10.1006/jdeq.1994.1049. |
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