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.
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, DOI 10.1016/j.jde.2007.11.011
H. Dulac, Sur les cycles limites, Bull. Soc. Math. France 51 (1923), 45–188 (French). MR 1504823
Freddy Dumortier, Compactification and desingularization of spaces of polynomial Liénard equations, J. Differential Equations 224 (2006), no. 2, 296–313. MR 2223719, DOI 10.1016/j.jde.2005.08.011
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, DOI 10.1090/S0002-9947-1991-1049864-0
Freddy Dumortier and Chengzhi Li, Quadratic Liénard equations with quadratic damping, J. Differential Equations 139 (1997), no. 1, 41–59. MR 1467352, DOI 10.1006/jdeq.1997.3291
Freddy Dumortier, Jaume Llibre, and Joan C. Artés, Qualitative theory of planar differential systems, Universitext, Springer-Verlag, Berlin, 2006. MR 2256001
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, DOI 10.1090/S0002-9939-07-08688-1
Freddy Dumortier and Robert Roussarie, Abelian integrals and limit cycles, J. Differential Equations 227 (2006), no. 1, 116–165. MR 2233957, DOI 10.1016/j.jde.2005.08.015
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, DOI 10.1006/jdeq.1994.1061
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
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, DOI 10.1090/S0273-0979-00-00881-8
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, DOI 10.1090/mmono/094
Yu. Ilyashenko, Centennial history of Hilbert’s 16th problem, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 301–354. MR 1898209, DOI 10.1090/S0273-0979-02-00946-1
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, DOI 10.17323/1609-4514-2001-1-4-583-599
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) Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977, pp. 335–357. MR 0448423
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, DOI 10.1016/j.jde.2003.10.008
Robert Roussarie, Bifurcation of planar vector fields and Hilbert’s sixteenth problem, Progress in Mathematics, vol. 164, Birkhäuser Verlag, Basel, 1998. MR 1628014, DOI 10.1007/978-3-0348-8798-4
Robert Roussarie, Putting a boundary to the space of Liénard equations, Discrete Contin. Dyn. Syst. 17 (2007), no. 2, 441–448. MR 2257444, DOI 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.
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
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
- [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 th problem for classical Liénard equations of even degree, J. Differential Equations 244 (2008), no. 6, 1359-1394. MR 2396502
- [Dul]
- H. Dulac, Sur les cycles limites, Bull. Soc. Math. France 51 (1923), 45-188. MR 1504823
- [D]
- F. Dumortier, Compactification and desingularization of spaces of polynomial Liénard equations, J. Differential Equations 224 (2006), no. 2, 296-313. MR 2223719
- [DF]
- F. Dumortier and P. Fiddelaers, Quadratic models for generic local -parameter bifurcations on the plane, Trans. Amer. Math. Soc. 326 (1991), no. 1, 101-126. MR 1049864
- [DL]
- F. Dumortier and C. Li, Quadratic Liénard equations with quadratic damping, J. Differential Equations 139 (1997), no. 1, 41-59. MR 1467352
- [DLA]
- F. Dumortier, J. Llibre, and J. C. Artés, Qualitative theory of planar differential systems, Universitext. Springer-Verlag, Berlin, 2006. xvi+298 pp., ISBN: 3-540-32893-9 MR 2256001
- [DPR]
- F. Dumortier, D. Panazzolo, and R. Roussarie, More limit cycles than expected in Liénard equations, Proc. Amer. Math. Soc. 135 (2007), 1895-1904. MR 2286102
- [DR]
- F. Dumortier and R. Roussarie, Abelian integrals and limit cycles, J. Differential Equations 227 (2006), 116-165. MR 2233957
- [DRR]
- F. Dumortier, R. Roussarie, and C. Rousseau, Hilbert's th problem for quadratic vector fields, J. Differential Equations 110 (1994), 86-133. MR 1275749
- [E]
- J. Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualitiées Math., Hermann, Paris, 1992. MR 1399559
- [H]
- D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc. 8 (1902), 437-479; reprinted, Bull. Amer. Math. Soc. (N.S.) 37 (2000), 407-436. MR 1779412
- [I1]
- Yu. Ilyashenko, Finiteness theorems for limit cycles, American Mathematical Society, Providence, RI, 1991. MR 1133882
- [I2]
- Yu. Ilyashenko, Centennial history of Hilbert's th problem, Bull. Amer. Math. Soc. (N.S.) 39 (2002), 301-354. MR 1898209
- [IP]
- Yu. Ilyashenko and A. Panov, Some upper estimates of the number of limit cycles of planar vector fields with applications to the Liénard equation, Mosc. Math. J. 1 (2001), no. 4, 583-599, 645. MR 1901077
- [LMP]
- A. Lins, W. De Melo, and C. C. Pugh, On Liénard's equation, Lectures Notes in Math. 597 (1977), 335-357. MR 0448423
- [LR]
- J. Llibre and G. Rodríguez, Configurations of limit cycles and planar polynomial vector fields, J. Differential Equations 198 (2004), no. 2, 374-380. MR 2039147
- [R1]
- R. Roussarie, Bifurcation of planar vector fields and Hilbert's th problem, Progress in Mathematics, Vol. 164, Birkhauser Verlag, Basel, 1998. MR 1628014
- [R2]
- R. Roussarie, Putting a boundary to the space of Liénard equations, Discrete Contin. Dyn. Syst. 17 (2007), no. 2, 441-448. MR 2257444
- [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]
- S. Shi, A concrete example of a quadratic system of the existence of four limit cycles for plane quadratic systems, Sci. Sinica 11 (1979), 1051-1056 (Chinese); 23 (1980), 153-158 (English). MR 0574405
- [Sm]
- S. Smale, Mathematical problems for the next century, Mathematical Intelligencer 20 (1998), no. 2, 7-15. MR 1819266
Review Information:
Reviewer:
Freddy Dumortier
Affiliation:
Hasselt University
Email:
freddy.dumortier@uhasselt.be
Journal:
Bull. Amer. Math. Soc.
46 (2009), 697-701
DOI:
https://doi.org/10.1090/S0273-0979-09-01267-1
Published electronically:
June 24, 2009
Review copyright:
© Copyright 2009
American Mathematical Society