| Citation: |
Xianbo Sun, Pei Yu. ZERO BIFURCATION DIAGRAMS FOR ABELIAN INTEGRALS: A STUDY ON HIGHER-ORDER HYPERELLIPTIC HAMILTONIAN SYSTEMS WITH THREE PERTURBATION PARAMETERS[J]. Journal of Applied Analysis & Computation, 2020, 10(6): 2734-2755. doi: 10.11948/20200168
|
ZERO BIFURCATION DIAGRAMS FOR ABELIAN INTEGRALS: A STUDY ON HIGHER-ORDER HYPERELLIPTIC HAMILTONIAN SYSTEMS WITH THREE PERTURBATION PARAMETERS
-
Abstract
In this paper, we present the zero bifurcation diagrams for the Abelian integrals of two hyperelliptic Hamiltonian systems with three perturbation parameters using an algebraic-geometric approach. The method can be used to study the bifurcation diagrams for higher-order Hamiltonian systems with polynomial perturbations of any degree. -
-
References
[1] V. I. Arnold, Ten problems, Adv. Soviet Math., 1990, 1, 1-8. [2] R. Asheghi and H. R. Zangeneh, Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop, Nonlinear Anal. (TMA), 2008, 68, 2957-2976. doi: 10.1016/j.na.2007.02.039 [3] R. Asheghi and H. R. Zangeneh, Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop (Ⅱ), Nonlinear Anal. (TMA), 2008, 69, 4143-4162. doi: 10.1016/j.na.2007.10.054 [4] R. Asheghi and H. R. Zangeneh, Bifurcations of limit cycles for a quintic Hamiltonian system with a double cuspidal loop, Comput. Math. Appl., 59(2010), 1409-1418. doi: 10.1016/j.camwa.2009.12.024 [5] M. Cndido, J. Llibre and C. Valls, Non-existence, existence, and uniqueness of limit cycles for a generalization of the Van der PolDuffing and the Rayleigh-Duffing oscillators, Physica D: Nonlinear Phenomena, 2020, 132458. [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, 114-157. doi: 10.1006/jdeq.2000.3977 [7] F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergodic theory and dynamical systems, 1987, 7, 375-413. doi: 10.1017/S0143385700004119 [8] V. Gathen and J. Gerhard, Modern Computer Algebra, Cambridge University Press, 2013. [9] M. Grau, F. Mañosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 2011, 363, 109-129. doi: 10.1090/S0002-9947-2010-05007-X [10] M. Gouveia, J. Llibre and L. A. Roberto, Phase portraits of the quadratic polynomial Liénarddifferential systems, Proc. Royal Soc. Edinb. A., 2020, 1-15. [11] M. Han, Asymptotic expansions of Abelian integrals and limit cycle bifurcations, Int. J. Bifur. Chaos, 2012, 22, 1250296 (30 pages). doi: 10.1142/S0218127412502963 [12] M. Han, J. Yang and D. Xiao, Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle, Int. J. Bifur. Chaos, 2012, 22, 1250189 (33 pages). doi: 10.1142/S0218127412501891 [13] M. Han, J. Yang and P. Yu, Hopf bifurcations for near-Hamiltonian systems, Int. J. Bifur. Chaos, 2009, 19, 4117-4130. doi: 10.1142/S0218127409025250 [14] M. Han and P. Yu, Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles, Springer, 2012. [15] M. Han, H. Zang and J. Yang, Limit cycle bifurcations by perturbing a cuspidal loop in a Hamiltonian system, J. Differ. Equat., 2009, 246, 129-163. doi: 10.1016/j.jde.2008.06.039 [16] M. Han, J. Yang, A. A. Tarta and Y. Gao, Limit cycles near homoclinic and heteroclinic loops, J. Dynam. Differ. Equat., 2008, 20, 923-960. doi: 10.1007/s10884-008-9108-3 [17] L. Hong, J. Lu and X. Hong, On the number of zeros of Abelian integrals for a class of quadratic reversible centers of genus one J. Nonlinear Modeling and Analysis, 2020, 2(2), 161-171. [18] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory (2nd Ed), Springer, New York, 1998. [19] C. Li and C. Rousseau, A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: The cusp of order 4, J. Differ. Equat., 1989, 79, 132-167. doi: 10.1016/0022-0396(89)90117-4 [20] C. Li and Z. Zhang, A criterion for determining the monotonicity of the ratio of two Abelian integrals, J. Differ. Equat., 1996, 124, 407-424. doi: 10.1006/jdeq.1996.0017 [21] J. Llibre, A. C. Murza and C. Valls, On a conjecture on the integrability of Liénard systems, Rendiconti del Circolo Matematico di Palermo Series 2, 2020, 69, 209-216. doi: 10.1007/s12215-018-00398-6 [22] F. Mañosas and J. Villadelprat, Bounding the number of zeros of certain Abelian integrals, J. Differ. Equat., 2011, 251, 1656-1669. doi: 10.1016/j.jde.2011.05.026 [23] P. Mardesić, D. Novikov, L. Ortiz-Bobadilla and J. Pontigo-Herrera, Infinite orbit depth and length of Melnikov functions, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2019, 36, 1941-1957. doi: 10.1016/j.anihpc.2019.07.003 [24] D. Novikov and S. Malev, Linear Estimate for the Number of Zeros of Abelian Integrals, Qual. The. Dynam. Syst., 2017, 16, 689-696. doi: 10.1007/s12346-016-0213-0 [25] L. Sheng, M. Han and Y. Tian, On the Number of Limit Cycles Bifurcating from a Compound Polycycle, Inter. J. Bifur. Chaos, 2020, 30, 2050099. doi: 10.1142/S0218127420500996 [26] X. Sun, M. Han and J. Yang, Bifurcation of limit cycles from a heteroclinic loop with a cusp, Nonlinear Anal. (TMA), 2011, 74, 2948-2965. doi: 10.1016/j.na.2011.01.013 [27] X. Sun and J. Yang, Sharp bounds of the number of zeros of Abelian integrals with parameters, E. J. Differ. Equat., 2014, 40, 1-12. [28] D. Xiao, Bifurcations on a five-parameter family of planar vector field, J. Dynam. Differ. Equat., 2008, 20, 961-980. doi: 10.1007/s10884-008-9109-2 [29] Y. Xiong and M. Han, A Note on the Expansion of the First Order Melnikov Function Near a Class of 3-polycycle, J. Nonl. Model. Anal., 2020, 2, 125-130. [30] J. Yang and X. Sun, Bifurcation of limit cycles for some Lienard systems with a nilpotent singular point, Inter. J. Bifur. Chaos, 2015, 25, 1550066(14pages), doi: 10.1142/S0218127415500662 [31] J. Yang and F. Liang, Limit cycle bifurcations of a kind of Lienard system with a hyperbolic saddle and a nilpotent cusp, J. Appl. Anal. Compu., 2015, 5:3, 515-526. [32] L. Zhao and D. Li, Bifurcations of limit cycles from a quintic Hamiltonian system with a heteroclinic cycle, Acta Math. Sinica, 2014, 30, 411-422. doi: 10.1007/s10114-014-2615-8 -
-
Figures(5) / Tables(1)
Export File
Citation
Xianbo Sun, Pei Yu. ZERO BIFURCATION DIAGRAMS FOR ABELIAN INTEGRALS: A STUDY ON HIGHER-ORDER HYPERELLIPTIC HAMILTONIAN SYSTEMS WITH THREE PERTURBATION PARAMETERS[J]. Journal of Applied Analysis & Computation, 2020, 10(6): 2734-2755. doi: 10.11948/20200168
Format
Content
DownLoad:
-
Figure 1. Phase portraits of system (1.2) showing (a) a nilpotent-saddle loop for
$ \alpha = \beta = 1 $ ; and (b) a cusp-saddle cycle for$ \alpha = 0 $ ,$ \beta = \frac{3}{5} $ . - Figure 2. Phase portrait of system (1.2) showing a family of closed orbits.
-
Figure 3. Bifurcation diagram of system (1.2) for
$ \alpha = \beta = 1 $ with$ \eta = \frac{a_0}{a_2} $ and$ \lambda = \frac{a_1}{a_2} $ , showing the Hopf bifurcation line$ L_1 $ , the homoclinic bifurcation line$ L_2 $ , the degenerate bifurcation points DL, DH and HL, and the curve C characterized by one zero of multiplicity two. -
Figure 4. Bifurcation diagram of system (1.2) for
$ \alpha = 0 $ and$ \beta = \frac{3}{5} $ with$ \eta = \frac{a_0}{a_2} $ and$ \lambda = \frac{a_1}{a_2} $ , showing the Hopf bifurcation line$ L_1 $ , the heteroclinic bifurcation line$ L_2 $ , the degenerate bifurcation points DC, DH and HC, and the curve C characterized by one zero of multiplicity two. -
Figure 5. The zero bifurcation diagram of
$ \oint_{\Gamma_h}(a_0+a_5 x^5+a_7x^7) y d x $ with$ \eta = \frac{a_0}{a_7} $ and$ \lambda = \frac{a_5}{a_7} $ , showing the Hopf bifurcation line$ L_1\!: \, \eta = 0 $ and the homoclinic bifurcation line$ L_2\!:\, \eta+\frac{497\lambda}{29172}+\frac{117}{14212} = 0 $ , and the degenerate bifurcation points HL$ \, = (-\frac{4563}{9443}, 0) $ , DL$ \, = (-\frac{109941}{108965}, \frac{976}{108965}) $ and DH$ \, = (0, 0) $ . Zero bifurcation in the regions is as follows: one zero in$ V_1 $ and$ V_3 $ , one zero of multiplicity two on C, two zeros in$ V_5 $ , and no zeros in$ V_2 $ and$ V_4 $ .