MONOTONICITY OF THE RATIO OF TWO ABELIAN INTEGRALS FOR A CLASS OF SYMMETRIC HYPERELLIPTIC HAMILTONIAN SYSTEMS

Rasool Kazemi

doi:10.11948/2018.344

2018 Volume 8 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2018 > 8(1): 344-355

Next Article Previous Article

Rasool Kazemi. MONOTONICITY OF THE RATIO OF TWO ABELIAN INTEGRALS FOR A CLASS OF SYMMETRIC HYPERELLIPTIC HAMILTONIAN SYSTEMS[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 344-355. doi: 10.11948/2018.344

Citation:

Rasool Kazemi. MONOTONICITY OF THE RATIO OF TWO ABELIAN INTEGRALS FOR A CLASS OF SYMMETRIC HYPERELLIPTIC HAMILTONIAN SYSTEMS[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 344-355. doi: 10.11948/2018.344

MONOTONICITY OF THE RATIO OF TWO ABELIAN INTEGRALS FOR A CLASS OF SYMMETRIC HYPERELLIPTIC HAMILTONIAN SYSTEMS

Rasool Kazemi

Department of Mathematical Sciences, University of Kashan, Km.;
6 Ravand road, 87317-53153, Kashan, Iran

Fund Project:

Abstract

Abstract

In this paper we study the monotonicity of the ratio of two hyperelliptic Abelian integrals I₀(h)=∮_Γh ydx and I₁(h)=∮_Γh xydx for which Γ_h is a continuous family of periodic orbits of a Newtonian system with Hamiltonian function of the form H(x,y)=1/2 y² ±Ψ(x), where Ψ is an arbitrary even function of degree six.
- Hamiltonian system /
- hyperelliptic Abelian integral /
- monotonicity /
- asymptotic expansion
MSC: 34C07;34C08;37G15;34M50

FullText(HTML)

References

[1]	C. Christopher and C. Li, Limit cycles of Differential Equations, Birkhäuser Verlag, 2007. Google Scholar
[2]	M. Grau, F. Ma~nosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 2011, 363, 109-129. Google Scholar
[3]	M. Han, Bifurcation Theory of Limit Cycles, Science Press, Beijing, 2013. Google Scholar
[4]	M. Han, J. Yang, A. Tarta and Y. Gao, Limit cycles near homoclinic and heteroclinic loops, Journal of Dynamics and Differential Equations, 2008, 20(4), 923-944. Google Scholar
[5]	C. Li and Z. F. Zhang, A criterion for determining the monotonicity of the ratio of two Abelian integrals, Journal of Differential Equations, 1996, 124(2), 407-424. Google Scholar
[6]	C. Liu and D. Xiao, The monotonicity of the ratio of two Abelian integrals, Transactions of the American Mathematical Society, 2013, 365(10), 5525-5544. Google Scholar
[7]	Y. Tian and M. Han, Hopf and homoclinic bifurcations for near-Hamiltonian systems, Journal of Differential Equations, 2017, 262(4), 3214-3234. Google Scholar
[8]	G. Tigan, Using Melnikov functions of any order for studying limit cycles, Journal of Mathematical Analysis and Applications, 2017, 448(1), 409-420. Google Scholar
[9]	N. Wang, D. Xiao and J. Yu, The monotonicity of the ratio of hyperelliptic integrals, Bulletin des Sciences Mathmatiques, 2014, 138(7), 805-845. Google Scholar