MONOTONICITY OF THE RATIOS OF TWO ABELIAN INTEGRALS FOR HAMILTONIAN SYSTEMS WITH PARAMETERS

Qiaoyun Wang; Na Wang; Xianbo Sun

doi:10.11948/20220349

2024 Volume 14 Issue 5

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Article navigation > Journal of Applied Analysis & Computation > 2024 > 14(5): 2466-2487

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Qiaoyun Wang, Na Wang, Xianbo Sun. MONOTONICITY OF THE RATIOS OF TWO ABELIAN INTEGRALS FOR HAMILTONIAN SYSTEMS WITH PARAMETERS[J]. Journal of Applied Analysis & Computation, 2024, 14(5): 2466-2487. doi: 10.11948/20220349

Citation:

Qiaoyun Wang, Na Wang, Xianbo Sun. MONOTONICITY OF THE RATIOS OF TWO ABELIAN INTEGRALS FOR HAMILTONIAN SYSTEMS WITH PARAMETERS[J]. Journal of Applied Analysis & Computation, 2024, 14(5): 2466-2487. doi: 10.11948/20220349

MONOTONICITY OF THE RATIOS OF TWO ABELIAN INTEGRALS FOR HAMILTONIAN SYSTEMS WITH PARAMETERS

1.
Department of Mathematics, Guangxi University, Nanning, Guangxi 530004, China
2.
School of Applied Sceince, Beijing Information Science and Technology, Beijing 100192, China
3.
School of Mathematics, Hangzhou Normal Univeristy, Hangzhou, Zhejiang 311121, China

Author Bio: Email: qiaoyunwang28@163.com(Q. Wang); Email: wn_math@126.com(N. Wang)
Corresponding author: Email: xsun@hznu.edu.cn(X. Sun)
Fund Project: This work is supported by Natural Science Foundation of Guangxi Province (2020JJG110003), and National Natural Science Foundation of China (Nos. 12001121, 11601385)

Abstract

Abstract

We study the monotonicity of the ratios of two Abelian integrals $ \oint_{\gamma_{i}(h)}ydx $ $ \backslash $ $ \oint_{\gamma_{i}(h)}xydx $ over three period annuli $ \{\gamma_i(h)\} $, for $ i=1, 2, 3 $, defined by a seventh-degree hyperelliptic Hamiltonian $ H(x, y)=y^2+\Psi(x) $ with a parameter. The parameter makes the problem more challenging to analyze. To overcome the difficulty, we apply some criterion with the help of transformations, tools in computer algebra such as boundary polynomial theory to determine the monotonicity of the ratios. Our results establish the existence and uniqueness of limit cycle bifurcated from each period annulus.
- Abelian integral /
- monotonicity /
- Hamiltonian system
MSC: 34C07, 34D10, 37G20

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References

[1]	V. I. Arnold, Ten Problems, Adv. Soviet. Math., 1990. Google Scholar
[2]	R. Asheghi and A. Bakhshalizadeh, The Chebyshev's property of certain hyperelliptic integrals of the first kind, Chaos, Solitons and Fractals, 2015, 78, 162–175. doi: 10.1016/j.chaos.2015.07.020 CrossRef Google Scholar
[3]	A. Bakhshalizadeh, R. Asheghi and R. Hoseyni, Zeros of hyperelliptic integrals of the first kind for special hyperelliptic Hamiltonians of degree 7, Chaos, Solitons and Fractals, 2017, 103, 279–288. doi: 10.1016/j.chaos.2017.06.021 CrossRef Google Scholar
[4]	A. Bakhshalizadeh, R. Asheghi and R. Kazemi, On the monotonicity of the ratio of some hyperelliptic integrals of order 7, Bull. Sci. Math., 2020, 158, 1–24. Google Scholar
[5]	R. Cheng, Z. Luo and X. Hong, Bifurcations and new traveling wave solutions for the nonlinear dispersion drinfel'd-Sokolov $D(m, n)$ system, J. Nonl. Mod. Anal., 2021, 3(2), 193–207. $D(m, n)$ system" target="_blank">Google Scholar
[6]	F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅰ) Saddle loop and two saddle cycle, J. Differ. Equat., 2001, 176(1), 114–157. doi: 10.1006/jdeq.2000.3977 CrossRef Google Scholar
[7]	F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅱ) Cuspidal loop, J. Differ. Equat., 2001, 175(2), 209–243. doi: 10.1006/jdeq.2000.3978 CrossRef Google Scholar
[8]	F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅲ) Global centre, J. Differ. Equat., 2003, 188(2), 473–511. doi: 10.1016/S0022-0396(02)00110-9 CrossRef Google Scholar
[9]	F. Dumortier and C. Li, Perturbations from an elliptic Hamiltonian of degree four: (Ⅳ) Figure eightloop, J. Differ. Equat., 2003, 188(2), 512–514. doi: 10.1016/S0022-0396(02)00111-0 CrossRef Google Scholar
[10]	A. Gasull, A. Geyer and V. Mañosa, Persistence of periodic traveling waves and Abelian integrals, J. Differ. Equat., 2021, 293, 48–69. doi: 10.1016/j.jde.2021.05.033 CrossRef Google Scholar
[11]	A. Geyer, R. Martins, F. Natali and D. Pelinovsky, Stability of smooth periodic travelling waves in the Camassa–Holm equation, Stud. Appl. Math., 2022, 148(1), 27–61. doi: 10.1111/sapm.12430 CrossRef Google Scholar
[12]	M. Grau, F. Mañosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 2011, 363(1), 109–129. Google Scholar
[13]	L. Guo and Y. Zhao, Existence of periodic waves for a perturbed quintic BBM equation, Disc. & Cont. Dynam. Syst., 2020, 40(8), 4689–4703. Google Scholar
[14]	M. Han and J. Yang, The maximum number of zeros of functions with parameters and application to differential equations, J. Nonl. Model. Anal., 2021, 3(1), 13–34. Google Scholar
[15]	R. Kazemi, Monotonicity of the ration of two Abelian integrals for a class of symmetric hyperelliptic hamiltonian systems, Bull. Sci. Math., 2018, 8(1), 344–355. Google Scholar
[16]	C. Li, J. Llibre and Z. Zhang, Abelian integrals of quadratic Hamiltonian vector field with an invariant straight line, Publications Matemàtiques., 1995, 39, 355–366. Google Scholar
[17]	C. Li and Z. Zhang, A criterion for determining the monotonicity of the ratio of two Abelian integrals, J. Differ. Equat., 1996, 124(2), 407–424. Google Scholar
[18]	F. Li, et al., Integrability and linearizability of cubic $Z_2$ systems with non-resonant singular points, J. Differ. Equat., 2020, 269(10), 9026–9049. $Z_2$ systems with non-resonant singular points" target="_blank">Google Scholar
[19]	F. Li, Y. Wu and P. Yu, Complete classification on center of cubic planar systems symmetric with respect to a straight line, Commun. Nonl. Sci. Numer. Simulat., 2023, 20, 107167. Google Scholar
[20]	C. Liu, G. Chen and Z. Sun, New criteria for the monotonicity of the ratio of two Abelian integrals, J. Math. Anal. Appl., 2018, 465(1), 220–234. Google Scholar
[21]	C. Liu and D. Xiao, The monotonicity of the ratio of two Abelian integrals, Trans. Amer. Math. Soc., 2013, 365(10), 5525–5544. Google Scholar
[22]	Y. Song, J. Shi and H. Wang, Spatiotemporal dynamics of a diffusive consumer-resource model with explicit spatial memory, Studies in Applied Mathematics, 2022, 148(1), 373–395. Google Scholar
[23]	X. Sun, W. Huang and J. Cai, Coexistence of the solitary and periodic waves in convecting shallow water fluid, Nonl. Anal. (RWA), 2020, 53, 1–17. Google Scholar
[24]	X. Sun, N. Wang and P. Yu, The monotonicity of ratios of some Abelian integrals, Bull. Sci. Math., 2021, 166, 2–11. Google Scholar
[25]	N. Wang, D. Xiao and J. Wang, The exact bounds on the number of zeros of complete hyperelliptic integrals of the first kind, J. Differ. Equat., 2013, 254(2), 323–341. Google Scholar
[26]	N. Wang, D. Xiao and J. Yu, The monotonicity of the ratio of hyperelliptic integrals, Bull. Sci. Math., 2014, 138(7), 805–845. Google Scholar
[27]	L. Yang, X. Hou and B. Xia, A complete algorithm for automated discovering of a class of inequalitytype theorems, Sci. China, Ser. F., 2001, 44(1), 33–49. Google Scholar
[28]	L. Yang and B. Xia, Real solution classification for parametric semi-algebraic systems, Algorithmic Algebra and Logic, 2005, 281–289. Google Scholar
[29]	Y. Zhou and J. Zhuang, Bifurcations and exact solutions of the Raman soliton model in nanoscale optical waveguides with metamaterials, J. Nonl. Mod. Anal., 2021, 3, 145–165. Google Scholar