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Book Review

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MathSciNet review: 2525744
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Book Information:

Authors: Colin Christopher and Chengzhi Li
Title: Limit cycles of differential equations
Additional book information: Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2007, viii+171 pp., ISBN 978-3-7643-8409-8

[BNY] B. Binyamini, D. Novikov, and S. Yakovenko, On the number of zeros of Abelian integrals: a constructive solution of the infinitesimal Hilbert th problem, ArXiv preprint 0808-2952 (2008), 1-57.
[CD] M. Caubergh and F. Dumortier, Hilbert’s 16th problem for classical Liénard equations of even degree, J. Differential Equations 244 (2008), no. 6, 1359–1394. MR 2396502, https://doi.org/10.1016/j.jde.2007.11.011
[Dul] H. Dulac, Sur les cycles limites, Bull. Soc. Math. France 51 (1923), 45–188 (French). MR 1504823
[D] Freddy Dumortier, Compactification and desingularization of spaces of polynomial Liénard equations, J. Differential Equations 224 (2006), no. 2, 296–313. MR 2223719, https://doi.org/10.1016/j.jde.2005.08.011
[DF] Freddy Dumortier and Peter Fiddelaers, Quadratic models for generic local 3-parameter bifurcations on the plane, Trans. Amer. Math. Soc. 326 (1991), no. 1, 101–126. MR 1049864, https://doi.org/10.1090/S0002-9947-1991-1049864-0
[DL] Freddy Dumortier and Chengzhi Li, Quadratic Liénard equations with quadratic damping, J. Differential Equations 139 (1997), no. 1, 41–59. MR 1467352, https://doi.org/10.1006/jdeq.1997.3291
[DLA] Freddy Dumortier, Jaume Llibre, and Joan C. Artés, Qualitative theory of planar differential systems, Universitext, Springer-Verlag, Berlin, 2006. MR 2256001
[DPR] Freddy Dumortier, Daniel Panazzolo, and Robert Roussarie, More limit cycles than expected in Liénard equations, Proc. Amer. Math. Soc. 135 (2007), no. 6, 1895–1904. MR 2286102, https://doi.org/10.1090/S0002-9939-07-08688-1
[DR] Freddy Dumortier and Robert Roussarie, Abelian integrals and limit cycles, J. Differential Equations 227 (2006), no. 1, 116–165. MR 2233957, https://doi.org/10.1016/j.jde.2005.08.015
[DRR] F. Dumortier, R. Roussarie, and C. Rousseau, Hilbert’s 16th problem for quadratic vector fields, J. Differential Equations 110 (1994), no. 1, 86–133. MR 1275749, https://doi.org/10.1006/jdeq.1994.1061
[E] Jean Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1992 (French, with French summary). MR 1399559
[H] David Hilbert, Mathematical problems, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 4, 407–436. Reprinted from Bull. Amer. Math. Soc. 8 (1902), 437–479. MR 1779412, https://doi.org/10.1090/S0273-0979-00-00881-8
[I1] Yu. S. Il′yashenko, Finiteness theorems for limit cycles, Translations of Mathematical Monographs, vol. 94, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by H. H. McFaden. MR 1133882
[I2] Yu. Ilyashenko, Centennial history of Hilbert’s 16th problem, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 301–354. MR 1898209, https://doi.org/10.1090/S0273-0979-02-00946-1
[IP] Yu. Ilyashenko and A. Panov, Some upper estimates of the number of limit cycles of planar vector fields with applications to Liénard equations, Mosc. Math. J. 1 (2001), no. 4, 583–599, 645 (English, with English and Russian summaries). Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary. MR 1901077, https://doi.org/10.17323/1609-4514-2001-1-4-583-599
[LMP] A. Lins, W. de Melo, and C. C. Pugh, On Liénard’s equation, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Springer, Berlin, 1977, pp. 335–357. Lecture Notes in Math., Vol. 597. MR 0448423
[LR] Jaume Llibre and Gerardo Rodríguez, Configurations of limit cycles and planar polynomial vector fields, J. Differential Equations 198 (2004), no. 2, 374–380. MR 2039147, https://doi.org/10.1016/j.jde.2003.10.008
[R1] Robert Roussarie, Bifurcation of planar vector fields and Hilbert’s sixteenth problem, Progress in Mathematics, vol. 164, Birkhäuser Verlag, Basel, 1998. MR 1628014
[R2] Robert Roussarie, Putting a boundary to the space of Liénard equations, Discrete Contin. Dyn. Syst. 17 (2007), no. 2, 441–448. MR 2257444, https://doi.org/10.3934/dcds.2007.17.441
[RS] V. G. Romanovski and D. S. Shafer, The center and cyclicity problems, a computational algebra approach, Birkhäuser, Boston, Basel, Berlin, 2008, ISBN 978-0-8176-4726-1.
[S] Song Ling Shi, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sinica 23 (1980), no. 2, 153–158. MR 574405
[Sm] S. Smale, Mathematical problems for the next century, Gac. R. Soc. Mat. Esp. 3 (2000), no. 3, 413–434 (Spanish). Translated from Math. Intelligencer 20 (1998), no. 2, 7–15 [ MR1631413 (99h:01033)] by M. J. Alcón. MR 1819266

Review Information:

Reviewer: Freddy Dumortier
Affiliation: Hasselt University
Email: freddy.dumortier@uhasselt.be

Journal: Bull. Amer. Math. Soc. 46 (2009), 697-701
MSC (2000): Primary 34C05, 34C07
DOI: https://doi.org/10.1090/S0273-0979-09-01267-1
Published electronically: June 24, 2009
Review copyright: © Copyright 2009 American Mathematical Society