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 & 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.  Google Scholar

[2]

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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/.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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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

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

show all references

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.  Google Scholar

[2]

-, Centers with degenerate infinity and their commutators, J. Math. Anal. Appl., 278 (2003), 109-124. doi: 10.1016/S0022-247X(02)00625-X.  Google Scholar

[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. Google Scholar

[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/.  Google Scholar

[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.  Google Scholar

[6]

I. Bendixson, Sur les courbes définies par des équations différentielles (French), Acta Math., 24 (1901), 1-88. doi: 10.1007/BF02403068.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[10]

R. Conti, Uniformly Isochronous Centers of Polynomial Systems in $\mathbbR^2$, Lecture Notes in Pure and Appl. Math., 152, Dekker, New York, 1994, (Differential equations, dynamical systems, and control science, 21-31).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

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