Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

The monotonicity of the ratio of two Abelian integrals


Authors: Changjian Liu and Dongmei Xiao
Journal: Trans. Amer. Math. Soc. 365 (2013), 5525-5544
MSC (2010): Primary 34C07, 34C08; Secondary 37G15
DOI: https://doi.org/10.1090/S0002-9947-2013-05934-X
Published electronically: May 10, 2013
MathSciNet review: 3074381
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study the monotonicity of the ratio of two Abelian integrals

$\displaystyle I_0(h)=\int _{\Gamma _h}y\,dx\quad {\rm and}\quad I_1(h)=\int _{\Gamma _h}xy\,dx,$

where $ \Gamma _h$ is a compact component of the level set $ \{(x,y):\ y^2+\Psi (x)=h, \ h\in J\}$; here $ J$ is an open interval. We first give a new criterion for determining the monotonicity of the ratio of the above two Abelian integrals. Then using this new criterion, we obtain some new Hamiltonian functions $ H(x,y)$ so that the ratio of the associated two Abelian integrals is monotone. Especially when $ H(x,y)$ has the form $ y^2+P_5(x)$, we obtain the sufficient and necessary conditions that the ratio of two Abelian integrals is monotone, where $ P_5(x)$ is a polynomial of $ x$ with degree five.

References [Enhancements On Off] (What's this?)

  • 1. V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, 1983. MR 695786 (84d:58023)
  • 2. V. I. Arnold, Ten Problems, in: Theory of singularities and its applications, Advance in Soviet Mathematics, 1(1990), 1-8. MR 1089668 (91k:58001)
  • 3. A. Atabaigi and H.Z. Zangeneh, Bifurcation of limit cycles in small perturbations of a class of hyperelliptic Hamiltonian systems of degree $ 5$ with a cusp, J. Appl. Anal. Comp., 1 (2011), 299-313.
  • 4. G. Binyamini, D. Novikov and S. Yakovenko, On the number of zeros of Abelian integrals, Invent. Math., 181 (2010), 227-289. MR 2657426 (2011m:34089)
  • 5. R. I. Bogdanov, Versal deformation of a singularity of a vector field on the plane in case of zero eigenvalues, Selecta Math. Soviet., 1 (1981), 389-421.
  • 6. F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four I, Saddle loop and two saddle cycle, J. Diff. Eqns., 176 (2001), 114-157. MR 1861185 (2002h:34061)
  • 7. F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four II, Cuspidal loop, J. Diff. Eqns., 175 (2001), 209-243. MR 1855970 (2002h:34060)
  • 8. F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four III, Global centre, J. Diff. Eqns., 188 (2001), 473-511. MR 1954291 (2003j:34046a)
  • 9. F. Dumortier and C. Li, Perturbation from an elliptic Hamiltonian of degree four IV. Figure eight-loop, J. Diff. Eqns., 188 (2003), 512-554. MR 1954292 (2003j:34046b)
  • 10. L. Gavrilov and I.D. Iliev, Completle hyperelliptic integrals of the first kind and their non-oscillation, Trans. Amer. Math. Soc., 356 (2003), 1185-1207. MR 2021617 (2004i:34077)
  • 11. M. Grau, F. Mañosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129. MR 2719674 (2011j:34096)
  • 12. R. Kazemi, H.Z. Zangeneh and A. Atabaigi, On the number of limit cycles in small perturbations of a class of hyperelliptic Hamiltonian systems, Nonlinear Analysis: Theory, Methods $ \&$ Appl., 76 (2012), 574-587.
  • 13. C. Li, Abelian Integrals and Limit Cycles, Qual. Theory Dyn. Syst., 11 (2012), 111-128. MR 2902727
  • 14. C. Li and Z. Zhang, A criterion for determining the monotonocity of the ratio of two Abelian integrals, J. Diff. Eqns., 124 (1996), 407-424. MR 1370149 (96i:34057)
  • 15. J. Li, Two results on Lienard equations, Ph.D. Thesis, Peking University, 1998.
  • 16. C. Liu, Estimate of the Number of Zeros of Abelian Integral for Elliptic Hamiltonian with Figure Eight-Loop, Nonlinearity, 16 (2003), 1151-1163. MR 1975800 (2003m:34077)
  • 17. D. Novikov and S. Yakovenko, Tangential Hilbert Problem for perturbations of Hyperelliptic Hamiltonian Systems, Electron. Announc. Amer. Math. Soc., 5 (1999), 55-65. MR 1679454 (2000a:34065)
  • 18. G. S. Petrov, Elliptic integrals and their nonoscillation (Russian), Funct. Anal. Appl., 20 (1986), no. 1, 46-49. MR 831048 (87f:58031)
  • 19. G. S. Petrov, Complex zeros of an elliptic integral, Func. Anal. Appl., 23 (1989), 88-89. MR 1011373 (90m:33002)
  • 20. G. S. Petrov, Nonoscillation of elliptic integrals, Func. Anal. Appl., 24 (1990), 205-210. MR 1082030 (92c:33036)
  • 21. G. S. Petrov, On the nonoscillation of elliptic integrals, Funct. Anal. Appl., 31 (1997), 262-265. MR 1608896 (99a:34087)
  • 22. R. Roussarie, On the number of limit cycles which appear by separatrix loop of planar vector fields, Bol. Soc. Bra. Math., 17 (1986), 67-101. MR 901596 (88i:34061)
  • 23. D. Shang and T. Zhang, Bifurcations of a cubic system perturbed by degree five, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 11-24. MR 2494080 (2010b:34064)
  • 24. J. Wang and D. Xiao, On the number of limit cycles in small perturbations of a class of hyperelliptic Hamiltonian systems with one nilpotent saddle, J. Diff. Equa., 250 (2011), 2227-2243. MR 2763571 (2012c:34087)
  • 25. T. Zhang, M. Tadé and Y. Tian, On the zeros of the Abelian integrals for a class of Liénard systems, Phys. Lett. A, 358 (2006), 262-274. MR 2253504 (2007g:34063)
  • 26. T. Zhang, Y. Tian and M. Tadé, On the number of zeros of the abelian integrals for a class of perturbed Liénard systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 3281-3287. MR 2372300 (2008k:34130)
  • 27. Y. Zhao and Zhifen Zhang, Linear estimate of the number of zeros of abelian integrals for a kind of quartic hamiltonians, J. Diff. Equa., 155 (1999), 73-88. MR 1693214 (2000g:34052)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 34C07, 34C08, 37G15

Retrieve articles in all journals with MSC (2010): 34C07, 34C08, 37G15


Additional Information

Changjian Liu
Affiliation: School of Mathematics, Soochow University, Suzhou 215006, People’s Republic of China
Email: liucj@suda.edu.cn

Dongmei Xiao
Affiliation: Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
Email: xiaodm@sjtu.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-2013-05934-X
Keywords: Abelian integral, monotonicity, hyperelliptic Hamiltonian
Received by editor(s): April 1, 2012
Published electronically: May 10, 2013
Additional Notes: The first author was partially supported by the NSFC grant (No. 10901117) and pre-research of Soochow University
The second author was the corresponding author and was partially supported by the NSFC grants (No. 10831003 and No. 10925102) and the Program of Shanghai Subject Chief Scientists (No. 10XD1406200)
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society