American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[8]

A. Garijo, A. Gasull and X. Jarque, Normal forms for singularities of one dimensional holomorphic vector fields,, \emph{Electron. J. Differential Equations}, 2004 ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Llibre, Integrability of polynomial differential systems, in Handbook of Differential Equations (Ordinary Differential Equations Volume I), Elsevier, Northholland, 2004, 437-532.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

H. Żoładek, Quadratic systems with center and their perturbations, J. Differential Equations, 109 (1994), 223-273. doi: 10.1006/jdeq.1994.1049.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[8]

A. Garijo, A. Gasull and X. Jarque, Normal forms for singularities of one dimensional holomorphic vector fields,, \emph{Electron. J. Differential Equations}, 2004 ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Llibre, Integrability of polynomial differential systems, in Handbook of Differential Equations (Ordinary Differential Equations Volume I), Elsevier, Northholland, 2004, 437-532.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

H. Żoładek, Quadratic systems with center and their perturbations, J. Differential Equations, 109 (1994), 223-273. doi: 10.1006/jdeq.1994.1049.  Google Scholar

[1]

Claudio A. Buzzi, Jeroen S.W. Lamb. Reversible Hamiltonian Liapunov center theorem. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 51-66. doi: 10.3934/dcdsb.2005.5.51
[2]

Andrey Gogolev, Ali Tahzibi. Center Lyapunov exponents in partially hyperbolic dynamics. Journal of Modern Dynamics, 2014, 8 (3&4) : 549-576. doi: 10.3934/jmd.2014.8.549
[3]

Vitali Kapovitch, Anton Petrunin, Wilderich Tuschmann. On the torsion in the center conjecture. Electronic Research Announcements, 2018, 25: 27-35. doi: 10.3934/era.2018.25.004
[4]

Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249
[5]

Keith Burns, Amie Wilkinson. Dynamical coherence and center bunching. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 89-100. doi: 10.3934/dcds.2008.22.89
[6]

Haihua Liang, Yulin Zhao. Quadratic perturbations of a class of quadratic reversible systems with one center. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 325-335. doi: 10.3934/dcds.2010.27.325
[7]

Iliya D. Iliev, Chengzhi Li, Jiang Yu. Bifurcations of limit cycles in a reversible quadratic system with a center, a saddle and two nodes. Communications on Pure & Applied Analysis, 2010, 9 (3) : 583-610. doi: 10.3934/cpaa.2010.9.583
[8]

B. Coll, A. Gasull, R. Prohens. Center-focus and isochronous center problems for discontinuous differential equations. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 609-624. doi: 10.3934/dcds.2000.6.609
[9]

Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839
[10]

Claudio Buzzi, Claudio Pessoa, Joan Torregrosa. Piecewise linear perturbations of a linear center. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 3915-3936. doi: 10.3934/dcds.2013.33.3915
[11]

Hongyu Cheng, Rafael de la Llave. Time dependent center manifold in PDEs. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6709-6745. doi: 10.3934/dcds.2020213
[12]

Rafael Ortega, Andrés Rivera. Global bifurcations from the center of mass in the Sitnikov problem. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 719-732. doi: 10.3934/dcdsb.2010.14.719
[13]

Jaume Llibre, Dana Schlomiuk. On the limit cycles bifurcating from an ellipse of a quadratic center. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1091-1102. doi: 10.3934/dcds.2015.35.1091
[14]

Samantha Erwin, Stanca M. Ciupe. Germinal center dynamics during acute and chronic infection. Mathematical Biosciences & Engineering, 2017, 14 (3) : 655-671. doi: 10.3934/mbe.2017037
[15]

Luis Barreira, Claudia Valls. Regularity of center manifolds under nonuniform hyperbolicity. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 55-76. doi: 10.3934/dcds.2011.30.55
[16]

Luis Barreira, Claudia Valls. Reversibility and equivariance in center manifolds of nonautonomous dynamics. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 677-699. doi: 10.3934/dcds.2007.18.677
[17]

Jaume Giné. Center conditions for generalized polynomial kukles systems. Communications on Pure & Applied Analysis, 2017, 16 (2) : 417-426. doi: 10.3934/cpaa.2017021
[18]

Martin Golubitsky, Claire Postlethwaite. Feed-forward networks, center manifolds, and forcing. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2913-2935. doi: 10.3934/dcds.2012.32.2913
[19]

Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873
[20]

Luis Barreira, Claudia Valls. Center manifolds for nonuniform trichotomies and arbitrary growth rates. Communications on Pure & Applied Analysis, 2010, 9 (3) : 643-654. doi: 10.3934/cpaa.2010.9.643

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (107)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences