LettersNo Access

ON THE NUMBER OF ZEROS OF THE ABELIAN INTEGRALS FOR A CLASS OF PERTURBED LIÉNARD SYSTEMS

Department of Chemical Engineering, Curtin University of Technology, GPO Box U1987, Perth WA 6845, Australia

Author for correspondence.

School of Software Engineering and Data Communications, Faculty of Information Technology, Queensland University of Technology, GPO Box 2434, Brisbane QLD 4001, Australia

Search for more papers by this author

, and

MOSES O. TADÉ

Department of Chemical Engineering, Curtin University of Technology, GPO Box U1987, Perth WA 6845, Australia

Search for more papers by this author

https://doi.org/10.1142/S0218127407019032Cited by:0 (Source: Crossref)

Abstract

Addressing the weakened Hilbert's 16th problem or the Hilbert–Arnold problem, this paper gives an upper bound B(n) ≤ 7n + 5 for the number of zeros of the Abelian integrals for a class of Liénard systems. We proved the main result using the Picard–Fuchs equations and the algebraic structure of the integrals.

Keywords:

References

V. I. Arnold, Adv. Soviet Math. 1, 1 (1990). Web of Science, Google Scholar
S. N. Chow, C. Li and Y. Yi, Erg. Th. Dyn. Syst. 22, 349 (2002). Web of Science, Google Scholar
F. Dumortier, R. Roussarie and C. Rousseau, J. Diff. Eqs. 110, 86 (1994), DOI: 10.1006/jdeq.1994.1061. Crossref, Web of Science, Google Scholar
Dumortier, F. & Li, C. [2001, 2003] "Perturbation from an elliptic Hamiltonian of degree four (I)–(III)," J. Diff. Eqs. 176, 114–157; 175, 209–243; 188, 473–511 . Google Scholar
L. Gavrilov, Bull. Lond. Math. Soc. 30, 267 (1998), DOI: 10.1112/S0024609397004244. Crossref, Web of Science, Google Scholar
L. Gavrilov, Invent. Math. 143, 449 (2001), DOI: 10.1007/PL00005798. Crossref, Web of Science, Google Scholar
H. Giacomini, J. Llibre and M. Viano, Nonlinearity 9, 50l (1996), DOI: 10.1088/0951-7715/9/2/013. Google Scholar
M. Han, Sci. China (Ser:A) 40, 1247 (1997), DOI: 10.1007/BF02876370. Crossref, Google Scholar
M. Han and J. Chen, Sci. China (Ser:A) 30, 401 (2000). Google Scholar
M. Han, T. Zhang and H. Zang, Int. J. Bifurcation and Chaos 14, 4285 (2004). Link, Web of Science, Google Scholar
E. Horozov and I. D. Iliev, Proc. Lond. Math. Soc. 69, 198 (1994). Web of Science, Google Scholar
E. Horozov and I. D. Ilive, Nonlinearity 11, 1521 (1998), DOI: 10.1088/0951-7715/11/6/006. Crossref, Web of Science, Google Scholar
I. D. Iliev, J. Lond. Math. Soc. 58, 353 (1998), DOI: 10.1112/S0024610798006486. Crossref, Web of Science, Google Scholar
Yu. Ilyashenko, Bull. Amer. Math. Soc. 39, 301 (2002), DOI: 10.1090/S0273-0979-02-00946-1. Crossref, Web of Science, Google Scholar
J. Li, Int. J. Bifurcation and Chaos 13, 47 (2003). Link, Web of Science, Google Scholar
Petrov, G. S. [1984–1990] "Number of zeros of complete elliptic integrals," Funct. Anal. Anal. 18 [1984] 72–73; 20 [1986] 37–40; 21 [1987] 87–88; 22 [1988] 37–40; 23, [1989] 88–89; 24, [1990] 45–50 . Google Scholar
R. Roussarie, Lecture Notes in Mathematics 1331 (Springer, NY, 1988) pp. 161–168. Google Scholar
S. Smale, Physica D 51, 261 (1991), DOI: 10.1016/0167-2789(91)90238-5. Google Scholar
H. Zang, T. Zhang and W. Chen, J. Compl. 20, 544 (2004). Google Scholar
T. Zhang, W. Chen and H. Zang, Int. J. Bifurcation and Chaos 14, 1853 (2004). Link, Web of Science, Google Scholar
T. Zhang, M. Han and H. Zang, Chaos Solit. Fract. 22, 1127 (2004), DOI: 10.1016/j.chaos.2004.03.028. Crossref, Web of Science, Google Scholar
T. Zhang, H. Zang and M. Han, Chaos Solit. Fract. 20, 629 (2004), DOI: 10.1016/j.chaos.2003.08.002. Crossref, Web of Science, Google Scholar
T. Zhang, M. Tadé and Y.-C. Tian, Chaos Solit. Fract. 31, 804 (2007), DOI: 10.1016/j.chaos.2005.10.029. Crossref, Web of Science, Google Scholar