# American Institute of Mathematical Sciences

March  2015, 35(3): 1075-1090. doi: 10.3934/dcds.2015.35.1075

## Center conditions for a class of planar rigid polynomial differential systems

 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia 2 Departamento de Ciencias, Pontificia Universidad Católica del Perú, Av. Universitaria 1801, San Miguel, Lima 32, Peru

Received  March 2014 Revised  August 2014 Published  October 2014

In general the center--focus problem cannot be solved, but in the case that the singularity has purely imaginary eigenvalues there are algorithms to solving it. The present paper implements one of these algorithms for the polynomial differential systems of the form $\dot x= -y + x f(x) g(y),\quad \dot y= x+y f(x) g(y),$ where $f(x)$ and $g(y)$ are arbitrary polynomials. These differential systems have constant angular speed and are also called rigid systems. More precisely, in this paper we give the center conditions for these systems, i.e. the necessary and sufficient conditions in order that they have an uniform isochronous center. In particular, the existence of a focus with the highest order is also studied.
Citation: Jaume Llibre, Roland Rabanal. Center conditions for a class of planar rigid polynomial differential systems. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1075-1090. doi: 10.3934/dcds.2015.35.1075
##### References:
 [1] A. Algaba and M. Reyes, Computing center conditions for vector fields with constant angular speed, J. Comput. Appl. Math., 154 (2003), 143-159. doi: 10.1016/S0377-0427(02)00818-X. [2] -, Centers with degenerate infinity and their commutators, J. Math. Anal. Appl., 278 (2003), 109-124. doi: 10.1016/S0022-247X(02)00625-X. [3] A. Algaba, M. Reyes, A. Bravo and T. Ortega, Campos cuárticos con velocidad angular constante, in Actas del XVI CEDYA '99, (1999), 1339-1340. [4] A. Algaba, M. Reyes and A. Bravo, Uniformly isochronous quintic planar vector fields, International Conference on Differential Equations, (Berlin, 1999), World Sci. Publ., River Edge, NJ, 1/2 (2000), 1415-1417. http://dynamics.mi.fu-berlin.de/equadiff/. [5] M. A. M. Alwash, On the center conditions of certain cubic systems, Proc. Amer. Math. Soc., 126 (1998), 3335-3336. doi: 10.1090/S0002-9939-98-04715-7. [6] I. Bendixson, Sur les courbes définies par des équations différentielles (French), Acta Math., 24 (1901), 1-88. doi: 10.1007/BF02403068. [7] J. Chavarriga, I. A. García and J. Giné, On integrability of differential equations defined by the sum of homogeneous vector fields with degenerate infinity, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 711-722. doi: 10.1142/S0218127401002390. [8] C. B. Collins, Algebraic conditions for a centre or a focus in some simple systems of arbitrary degree, J. Math. Anal. Appl., 195 (1995), 719-735. doi: 10.1006/jmaa.1995.1385. [9] -, Conditions for a centre in a simple class of cubic systems, Differential Integral Equations, 10 (1997), 333-356. http://projecteuclid.org/euclid.die/1367526341 [10] R. Conti, Uniformly Isochronous Centers of Polynomial Systems in $\mathbb{R}^2$, Lecture Notes in Pure and Appl. Math., 152, Dekker, New York, 1994, (Differential equations, dynamical systems, and control science, 21-31). [11] A. Cima, A. Gasull and F. Mañosas, Cyclicity of a family of vector fields, J. Math. Anal. Appl., 196 (1995), 921-937. doi: 10.1006/jmaa.1995.1451. [12] F. S. Dias and L. F. Mello, The center-focus problem and small amplitude limit cycles in rigid systems, Discrete Contin. Dyn. Syst., 32 (2012), 1627-1637. doi: 10.3934/dcds.2012.32.1627. [13] H. Dulac, Détermination et intégration d'une certaine classe d'quations diffŕentielles ayant pour point singulier un centre, Bull. Sci. Math Sér. (2), 32 (1908), 230-252. [14] W. W. Farr, Li. Chengzhi, I. S. Labouriau and W. F. Langford, Degenerate Hopf bifurcation formulas and Hilbert's 16th problem, SIAM J. Math. Anal., 20 (1989), 13-30. doi: 10.1137/0520002. [15] M. Frommer, Über das Auftreten von Wirbeln und Strudeln (geschlossener und spiraliger Integralkurven) in der Umgebung rationaler Unbestimmtheitsstellen, Math. Ann., 109 (1934), 395-424. doi: 10.1007/BF01449147. [16] 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. [17] A. Gasull and J. Torregrosa, Exact number of limit cycles for a family of rigid systems, Proc. Amer. Math. Soc., 133 (2005), 751-758. doi: 10.1090/S0002-9939-04-07542-2. [18] A. Gasull, R. Prohens and J. Torregrosa, Limit cycles for rigid cubic systems, J. Math. Anal. Appl., 303 (2005), 391-404. doi: 10.1016/j.jmaa.2004.07.030. [19] W. Kapteyn, On the midpoints of integral curves of differential equations of the first degree (Dutch), Nederl. Akad. Wetensch. Verslag Afd. Natuurk. Konikl. Nederland, 19 (1911), 1446-1447. [20] -, New investigations on the midpoints of integral curves of differential equations of the first degree (Dutch) Nederl. Akad. Wetensch. Verslag Afd. Natuurk. Konikl. Nederland, 20 (1912), 1354-1365; 21 (1912), 27-33. [21] A. M. Liapunov, Stability of Motion, Math. Sci. Engrg. 30 Academic Press, New York-London, 1966. xi+203 pp. doi: 10.1016/S0076-5392(08)61301-6. [22] -, Problème Général de la Stabilité du Mouvement, Annals of Mathematics Studies 17, Princeton University Press, Princeton, N. J.; Oxford University Press, London, 1947. iv+272 pp. [23] 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. [24] L. Mazzi and M. Sabatini, Commutators and linearizations of isochronous centers, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 11 (2000), 81-98. http://eudml.org/doc/252342 [25] 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. [26] H. Poincaré, Mémoire sur les courbes définies par une équation différentielle, Edit. Jacques Gabay, Paris, 1993. Reprinted from the original papers published in the Journal de Mathématiques, 7 (1881), 375-422, 8 (1882), 251-296, 1 (1885), 167-244, and 2 (1886), 151-217. [27] R. Roussarie, Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem, Progress in Mathematics 164. Birkhäuser Verlag, Basel, 1998. doi: 10.1007/978-3-0348-8798-4. [28] S. D. Shafer, Symbolic computation and the cyclicity problem for singularities, J. Symbolic Comput., 47 (2012), 1140-1153. doi: 10.1016/j.jsc.2011.12.037. [29] E. P. Volokitin, Center conditions for a simple class of quintic systems. Int. J. Math. Math. Sci., 29 (2002), 625-632. doi: 10.1155/S0161171202012802. [30] -, Centering conditions for planar septic systems, Electron. J. Differential Equations, 34 (2003), 1-7. http://ejde.math.txstate.edu/Volumes/2003/34/abstr.html

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##### References:
 [1] A. Algaba and M. Reyes, Computing center conditions for vector fields with constant angular speed, J. Comput. Appl. Math., 154 (2003), 143-159. doi: 10.1016/S0377-0427(02)00818-X. [2] -, Centers with degenerate infinity and their commutators, J. Math. Anal. Appl., 278 (2003), 109-124. doi: 10.1016/S0022-247X(02)00625-X. [3] A. Algaba, M. Reyes, A. Bravo and T. Ortega, Campos cuárticos con velocidad angular constante, in Actas del XVI CEDYA '99, (1999), 1339-1340. [4] A. Algaba, M. Reyes and A. Bravo, Uniformly isochronous quintic planar vector fields, International Conference on Differential Equations, (Berlin, 1999), World Sci. Publ., River Edge, NJ, 1/2 (2000), 1415-1417. http://dynamics.mi.fu-berlin.de/equadiff/. [5] M. A. M. Alwash, On the center conditions of certain cubic systems, Proc. Amer. Math. Soc., 126 (1998), 3335-3336. doi: 10.1090/S0002-9939-98-04715-7. [6] I. Bendixson, Sur les courbes définies par des équations différentielles (French), Acta Math., 24 (1901), 1-88. doi: 10.1007/BF02403068. [7] J. Chavarriga, I. A. García and J. Giné, On integrability of differential equations defined by the sum of homogeneous vector fields with degenerate infinity, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 711-722. doi: 10.1142/S0218127401002390. [8] C. B. Collins, Algebraic conditions for a centre or a focus in some simple systems of arbitrary degree, J. Math. Anal. Appl., 195 (1995), 719-735. doi: 10.1006/jmaa.1995.1385. [9] -, Conditions for a centre in a simple class of cubic systems, Differential Integral Equations, 10 (1997), 333-356. http://projecteuclid.org/euclid.die/1367526341 [10] R. Conti, Uniformly Isochronous Centers of Polynomial Systems in $\mathbb{R}^2$, Lecture Notes in Pure and Appl. Math., 152, Dekker, New York, 1994, (Differential equations, dynamical systems, and control science, 21-31). [11] A. Cima, A. Gasull and F. Mañosas, Cyclicity of a family of vector fields, J. Math. Anal. Appl., 196 (1995), 921-937. doi: 10.1006/jmaa.1995.1451. [12] F. S. Dias and L. F. Mello, The center-focus problem and small amplitude limit cycles in rigid systems, Discrete Contin. Dyn. Syst., 32 (2012), 1627-1637. doi: 10.3934/dcds.2012.32.1627. [13] H. Dulac, Détermination et intégration d'une certaine classe d'quations diffŕentielles ayant pour point singulier un centre, Bull. Sci. Math Sér. (2), 32 (1908), 230-252. [14] W. W. Farr, Li. Chengzhi, I. S. Labouriau and W. F. Langford, Degenerate Hopf bifurcation formulas and Hilbert's 16th problem, SIAM J. Math. Anal., 20 (1989), 13-30. doi: 10.1137/0520002. [15] M. Frommer, Über das Auftreten von Wirbeln und Strudeln (geschlossener und spiraliger Integralkurven) in der Umgebung rationaler Unbestimmtheitsstellen, Math. Ann., 109 (1934), 395-424. doi: 10.1007/BF01449147. [16] 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. [17] A. Gasull and J. Torregrosa, Exact number of limit cycles for a family of rigid systems, Proc. Amer. Math. Soc., 133 (2005), 751-758. doi: 10.1090/S0002-9939-04-07542-2. [18] A. Gasull, R. Prohens and J. Torregrosa, Limit cycles for rigid cubic systems, J. Math. Anal. Appl., 303 (2005), 391-404. doi: 10.1016/j.jmaa.2004.07.030. [19] W. Kapteyn, On the midpoints of integral curves of differential equations of the first degree (Dutch), Nederl. Akad. Wetensch. Verslag Afd. Natuurk. Konikl. Nederland, 19 (1911), 1446-1447. [20] -, New investigations on the midpoints of integral curves of differential equations of the first degree (Dutch) Nederl. Akad. Wetensch. Verslag Afd. Natuurk. Konikl. Nederland, 20 (1912), 1354-1365; 21 (1912), 27-33. [21] A. M. Liapunov, Stability of Motion, Math. Sci. Engrg. 30 Academic Press, New York-London, 1966. xi+203 pp. doi: 10.1016/S0076-5392(08)61301-6. [22] -, Problème Général de la Stabilité du Mouvement, Annals of Mathematics Studies 17, Princeton University Press, Princeton, N. J.; Oxford University Press, London, 1947. iv+272 pp. [23] 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. [24] L. Mazzi and M. Sabatini, Commutators and linearizations of isochronous centers, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 11 (2000), 81-98. http://eudml.org/doc/252342 [25] 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. [26] H. Poincaré, Mémoire sur les courbes définies par une équation différentielle, Edit. Jacques Gabay, Paris, 1993. Reprinted from the original papers published in the Journal de Mathématiques, 7 (1881), 375-422, 8 (1882), 251-296, 1 (1885), 167-244, and 2 (1886), 151-217. [27] R. Roussarie, Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem, Progress in Mathematics 164. Birkhäuser Verlag, Basel, 1998. doi: 10.1007/978-3-0348-8798-4. [28] S. D. Shafer, Symbolic computation and the cyclicity problem for singularities, J. Symbolic Comput., 47 (2012), 1140-1153. doi: 10.1016/j.jsc.2011.12.037. [29] E. P. Volokitin, Center conditions for a simple class of quintic systems. Int. J. Math. Math. Sci., 29 (2002), 625-632. doi: 10.1155/S0161171202012802. [30] -, Centering conditions for planar septic systems, Electron. J. Differential Equations, 34 (2003), 1-7. http://ejde.math.txstate.edu/Volumes/2003/34/abstr.html
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