Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2019, Article ID 6943563, 12 pages
https://doi.org/10.1155/2019/6943563
Research Article

Poincaré Bifurcation of Limit Cycles from a Liénard System with a Homoclinic Loop Passing through a Nilpotent Saddle

Department of Applied Mathematics, Guangxi University of Finance and Economics, Nanning, Guangxi, 530003, China

Correspondence should be addressed to Minzhi Wei; moc.361@321gnoixnayoaix

Received 22 February 2019; Accepted 14 May 2019; Published 2 June 2019

Academic Editor: Douglas R. Anderson

Copyright © 2019 Minzhi Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. Li, “Hilbert's 16th problem and bifurcations of planar polynomial vector fields,” International Journal of Bifurcation and Chaos, vol. 13, no. 1, pp. 47–106, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  2. Y. Ilyashenko, “Centennial history of Hilbert's 16th problem,” Bulletin of the American Mathematical Society, vol. 39, no. 3, pp. 301–354, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  3. D. Schlomiuk, “Algebraic and geometric aspects of the theory of polynomial vector fields,” in Bifurcations and Periodic Orbits of Vector Fields, D. Schlomiuk, Ed., vol. 408 of NATO ASI Series C, pp. 429–467, Kluwer Academic, London, UK, 1993. View at Google Scholar · View at MathSciNet
  4. L. S. Chen and M. S. Wang, “The relative position, and the number, of limit cycles of a quadratic differential system,” Acta Mathematica Sinica, vol. 22, no. 6, pp. 751–758, 1979. View at Google Scholar · View at MathSciNet
  5. S. L. Shi, “A concrete example of the existence of four limit cycles for plane quadratic systems,” Scientia Sinica, vol. 23, no. 2, pp. 153–158, 1980. View at Google Scholar · View at MathSciNet
  6. J. B. Li and C. F. Li, “Planar cubic Hamiltonian systems and distribution of limit cycles of (E3),” Acta Mathematica Sinica, vol. 28, no. 4, pp. 509–521, 1985. View at Google Scholar · View at MathSciNet
  7. M. Han, Y. Wu, and P. Bi, “A new cubic system having eleven limit cycles,” Discrete and Continuous Dynamical Systems - Series A, vol. 12, no. 4, pp. 675–686, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  8. H. Maoan, Z. Tonghua, and Z. Hong, “On the number and distribution of limit cycles in a cubic system,” International Journal of Bifurcation and Chaos, vol. 14, no. 12, pp. 4285–4292, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. C. Li, C. Liu, and J. Yang, “A cubic system with thirteen limit cycles,” Journal of Differential Equations, vol. 246, no. 9, pp. 3609–3619, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. M. Han, D. Shang, W. Zheng, and P. Yu, “Bifurcation of limit cycles in a fourth-order near-Hamiltonian system,” International Journal of Bifurcation and Chaos, vol. 17, no. 11, pp. 4117–4144, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  11. J. Li, H. S. Chan, and K. W. Chung, “Bifurcations of limit cycles in a -equivariant planar vector field of degree 5,” Science China Mathematics, vol. 45, no. 7, pp. 817–826, 2002. View at Google Scholar · View at MathSciNet
  12. S. Wang and P. Yu, “Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation,” Chaos, Solitons & Fractals, vol. 26, no. 5, pp. 1317–1335, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  13. S. Wang and P. Yu, “Existence of 121 limit cycles in a perturbed planar polynomial Hamiltonian vector field of degree 11,” Chaos, Solitons & Fractals, vol. 30, no. 3, pp. 606–621, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  14. T. Johnson and W. Tucker, “An improved lower bound on the number of limit cycles bifurcating from a Hamiltonian planar vector field of degree 7,” International Journal of Bifurcation and Chaos, vol. 20, no. 5, pp. 1451–1458, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  15. S. Wang, P. Yu, and J. Li, “Bifurcation of limit cycles in -equivariant vector fields of degree 9,” International Journal of Bifurcation and Chaos, vol. 16, no. 8, pp. 2309–2324, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  16. F. Dumortier and C. Li, “Perturbation from an elliptic Hamiltonian of degree four. (IV). Figure eight-loop,” Journal of Differential Equations, vol. 188, no. 2, pp. 512–554, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  17. H. Zhu, B. Qin, S. Yang, and M. Wei, “Poincare' bifurcation of some nonlinear oscillator of generalized lie'nard type using symbolic computation method,” International Journal of Bifurcation and Chaos, vol. 28, no. 8, Article ID 1850096, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  18. X. Sun and J. Yang, “Sharp bounds of the number of zeros of Abelian integrals with parameters,” Electronic Journal of Differential Equations, vol. 40, pp. 1–12, 2014. View at Google Scholar · View at MathSciNet
  19. L. Q. Zhao and D. P. Li, “Bifurcations of limit cycles from a quintic Hamiltonian system with a heteroclinic cycle,” Acta Mathematica Sinica, vol. 30, no. 3, pp. 411–422, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  20. X. Sun, “Perturbation of a period annulus bounded by a heteroclinic loop connecting two hyperbolic saddles,” Qualitative Theory of Dynamical Systems, vol. 16, no. 1, pp. 187–203, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. R. Asheghi and H. R. Zangeneh, “Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop (II),” Nonlinear Analysis, vol. 69, no. 11, pp. 4143–4162, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  22. J. Wang and D. Xiao, “On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle,” Journal of Differential Equations, vol. 250, no. 4, pp. 2227–2243, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. X. Sun, H. Xi, H. R. Zangeneh, and R. Kazemi, “Bifurcation of limit cycles in small perturbation of a class of Lie'nard systems,” International Journal of Bifurcation and Chaos, vol. 24, no. 1, Article ID 1450004, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  24. T. Zhang, M. O. Tade', and Y.-C. Tian, “On the zeros of the Abelian integrals for a class of Liénard systems,” Physics Letters A, vol. 358, no. 4, pp. 262–274, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  25. F. Dumortier and C. Li, “Perturbations from an elliptic Hamiltonian of degree four. I. Saddle loop and two saddle cycle,” Journal of Differential Equations, vol. 176, no. 1, pp. 114–157, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. F. Dumortier and C. Li, “Perturbations from an elliptic Hamiltonian of degree four. (II). Cuspidal loop,” Journal of Differential Equations, vol. 175, no. 2, pp. 209–243, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  27. F. Dumortier and C. Li, “Perturbation from an elliptic Hamiltonian of degree four. (III). Global centre,” Journal of Differential Equations, vol. 188, no. 2, pp. 473–511, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  28. L. Zhao, “The perturbations of a class of hyper-elliptic Hamilton systems with a double homoclinic loop through a nilpotent saddle,” Nonlinear Analysis, vol. 95, pp. 374–387, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  29. X. Sun and W. Huang, “Bounding the number of limit cycles for a polynomial Liénard system by using regular chains,” Journal of Symbolic Computation, vol. 79, no. part 2, pp. 197–210, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  30. X. Sun and L. Zhao, “Perturbations of a class of hyper-elliptic Hamiltonian systems of degree seven with nilpotent singular points,” Applied Mathematics and Computation, vol. 289, pp. 194–203, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. X. B. Sun and M. A. Han, “On the number of limit cycles of a -equivariant quintic near-Hamiltonian system,” Acta Mathematica Sinica, vol. 31, no. 11, pp. 1805–1824, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  32. X. Sun, P. Yu, and B. Qin, “lobal existence and uniqueness of periodic waves in a population model with density-dependent migrations and Allee effect,” International Journal of Bifurcation and Chaos, vol. 27, no. 12, Article ID 1750192, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  33. M. Grau, F. Maosas, and J. Villadelprat, “A Chebyshev criterion for Abelian integrals,” Transactions of the American Mathematical Society, vol. 363, pp. 109–129, 2011. View at Publisher · View at Google Scholar
  34. F. Maosas and J. Villadelprat, “Bounding the number of zeros of certain Abelian integrals,” Journal of Differential Equations, vol. 251, pp. 1656–1669, 2011. View at Publisher · View at Google Scholar
  35. M. Maza, “On triangular decompositions of algebraic varieties,” in Proceedings of the Mega-2000 Conference, Bath, UK, 2000.