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Perturbation of a Period Annulus Bounded by a Saddle–Saddle Cycle in a Hyperelliptic Hamiltonian Systems of Degree Seven

  • Yang, Sumin [1] ; Qin Bin [1] ; Xia Guoen [1] ; Yong-Hui, Xia [2]
    1. [1] Guangxi University of Finance and Economics

      Guangxi University of Finance and Economics

      China

    2. [2] Zhejiang Normal University

      Zhejiang Normal University

      China

  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
  • Idioma: inglés
  • DOI: 10.1007/s12346-020-00348-7
  • Enlaces
  • Resumen

    • In this paper, we study the limit cycle bifurcation by perturbing the period annuluses of two perturbed hyper-elliptic Hamiltonian systems of degree seven. The period annuluses are bounded by heteroclinic loops, inside or outside of which there exist two nilpotent cusps. The bifurcation function is Abelian integral which is the first-order approximation of the Poincaré map. The sharp bounds of the number of limit cycles bifurcated from the periodic annuluses are obtained by Chebyshev criterion and asymptotic analysis.

  • Referencias bibliográficas
    • 1. Hilbert, D.: Mathematical problems. Bull. Am. Math. Soc. 8, 437–479 (1902)
    • 2. Françoise, J.P.: Local bifurcations of limit cycles, Abel equations and Liénard systems. Normal Forms Bifurc. Finiteness Probl. Differ....
    • 3. Zhao, L., Fan, Z.: The number of small amplitude limit cycles in arbitrary polynomial systems. J. Math. Anal. Appl. 407, 237–249 (2013)
    • 4. Smale, S.: Mathematical problems for the next century. Math, Intell. 20, 7–15 (1998)
    • 5. Arnold, V.I.: Ten problems. Adv. Sov. Math. 1, 1–8 (1990)
    • 6. Han, M.: Bifurcation Theory of Limit Cycles. Science Press, Beijing (2013)
    • 7. Chen, F., Li, Chengzhi, Llibre, Jaume, Zhang, Zenghua: A unified proof on the weak Hilbert 16th problem for n = 2. J. Differ. Equ....
    • 8. Sun, X., Yu, P., Qin, B.: Global existence and uniqueness of periodic waves in a population model with density-dependent migrations and...
    • 9. Sun, X., Yu, P.: Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms. Discrete Cont....
    • 10. Wang, J., Xiao, D.: On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent...
    • 11. Wang, J.: Perturbations of Several Planar Integral System with Denerate Singularity. Shanghai Jiaotong Univesity, Shanghai (2012)
    • 12. Atabaigi, A., Zangeneh, R.: Bifurcation of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems of degree...
    • 13. Sun, X., Su, J., Han, M.: On the number of zeros of Abelian integral for some Liénard system of type (4,3). Chaos Solitons Fractals 51,...
    • 14. Sun, X., Wu, K.: The sharp bound on the number of zeros of Abelian integral for a perturbation of hyper-elliptic Hamiltonian system (in...
    • 15. Sun, X., Huang, W.: Bounding the number of limit cycles for a polynomial Liénard system by using regular chains. J. Symb. Comput. 79,...
    • 16. Asheghi, R., Zangeneh, H.R.Z.: Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-gure loop. Nonlinear Anal. 68,...
    • 17. Asheghi, R., Zangeneh, H.R.Z.: Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop (II). Nonlinear Anal....
    • 18. Asheghi, R., Zangeneh, R.: Bifurcations of limit cycles for a quintic Hamiltonian system with a double cuspidal loop. Comput. Math. Appl....
    • 19. Zhao, L., Qi, M., Liu, C.: The cyclicity of period annuli of a class of quintic Hamiltonian system. J. Math. Anal. Appl. 403, 391–407...
    • 20. Zhao, L.: The perturbations of a class of hyper-elliptic Hamilton systems with a double homoclinic loop through a nilpotent saddle. Nonlinear...
    • 21. Sun, X., Yang, J.: Sharp bounds of number of zeros of Abelian integral with parameters. Electron. J. Differ. Equ. 40, 1–12 (2014)
    • 22. Sun, X.: Bifurcation of limit cycles from a hyper-elliptic Hamiltonian system with a double heteroclinic loops. Adv. Differ. Equ. 2012,...
    • 23. Zhao, L., Li, D.: Bifurcations of limit cycles from a quintic Hamiltonian system with a heteroclinic cycle. Acta Math. Sin. (English Series)...
    • 24. Kazemi, R., Zangeneh, H.R.Z., Atabaigi, A.: On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian...
    • 25. Sun, X., Xi, H., Zangeneh, R., Kazemi, R.: Bifurcation of limit cycles in small perturbation of a class of Lie´nard systems. Int. J. Bifurc....
    • 26. Kazemi, R., Zangeneh, R.: Bifurcation of limit cycles in small perturbations of a hyper-elliptic Hamiltonian system with two nilpotent...
    • 27. Sun, X.: Bifurcation of limit cycles from a Liénard system with a heteroclinic loop connecting two nilpotent saddles. Nonlinear Dyn. 73,...
    • 28. Sun, X., Zhao, L.: Perturbations of a class of hyper-elliptic Hamiltonian systems of degree seven with nilpotent singular points. Appl....
    • 29. Sun, X.: Perturbation of a period annulus bounded by a heteroclinic loop connecting two hyperbolic saddles. Qual. Theory Dyn. Syst. 16,...
    • 30. Grau, M., Mañosas, F., Villadelprat, J.: A Chebyshev criterion for Abelian integrals. Trans. Am. Math. Soc. 363, 109–129 (2011)
    • 31. Mañosas, F., Villadelprat, J.: Bounding the number of zeros of certain Abelian integrals. J. Differ. Equ. 251, 1656–1669 (2011)
    • 32. Han, M.: Asymptotic expansions of Melnikov functions and limit cycle bifurcations. Int. J. Bifurc. Chaos 22, 1250296 (2012)
    • 33. Han, M., Sheng, L.: Bifurcation of limit cycles for piesewise smooth system via Melnikov function. J. Appl. Anal. Comput. 5, 809–815 (2015)
    • 34. Zhu, W., Xia, Y., Zhang, B., Bai, Y.Z.: Exact traveling wave solutions and bifurcations of the time fractional differential equations...
    • 35. Xia, Y., Grasic, M., Huang, W., Romanovski, V.G.: Limit cycles in a model of olfactory sensory neurons. Int. J. Bifurc. Chaos 29, 1950038...
    • 36. Han, M., Sheng, L., Zhang, X.: Bifurcation theory for finitely smooth planar autonomous differential systems. J. Differ. Equ. 264, 3596–3618...
    • 37. Han, M., Yang, J., Tarta, A., Gao, Y.: Limit cycles near homoclinic and heteroclinic loops. J. Dyn. Differ. Equ. 20, 923–944 (2008)
    • 38. Han, M., Yang, J., Yu, P.: Hopf bifurcations for near-Hamiltonian systems. Int. J. Bifurc. Chaos 19, 4117–4130 (2009)
    • 39. Sun, X., Han, M., Yang, J.: Bifurcation of limit cycles from a heteroclinic loop with a cusp. Nonlinear Anal. (TMA) 74, 2948–2965 (2011)
    • 40. Han, M., Wang, Z., Zang, H.: Limit cycles by Hopf and homoclinic bifurcations for near-Hamiltonian systems. Chin. J. Cont. Math. 28, 423–434...
    • 41. Chen, H., Han, M., Xia, Y.: Limit cycles of a Liénard system with symmetry allowing for discontinuity. J. Math. Anal. Appl. 468, 799–816...
    • 42. Tian, H., Han, M.: Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems. J. Differ. Equ....
    • 43. Yang, J., Zhao, L.: Bounding the number of limit cycles of discontinuous differential systems by using Picard–Fuchs equations. J. Differ....
    • 44. Llibre, J., Ponce, E., Valls, C.: Uniqueness and non-uniqueness of limit cycles for piecewise linear differential systems with three zones...
    • 45. Llibre, J., Teixeira, M., Torregrosa, J.: Lower bounds for the maximum number of limit cycles of discontinuous piecewise Llinear differential...
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