Abstract
The limit cycle bifurcation of a plane fifth-order vector field with double homoclinic polycyclic rings is studied by qualitative analysis and numerical exploration. This study is based on a detection function that is particularly effective for perturbed planar polynomial system. The research shows that a class of five-order vector field has 5 limit cycles under asymmetric disturbance. The asymmetric perturbation here has 4 arbitrary parameters. Using the numerical simulation method, the distributed orderliness of the 5 limit cycles is observed. This will help to further study Hilbert’s 16th question.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Dumortier, F., Li, C.: Perturbations from an elliptic Hamiltonian of degree four, saddle loop and two saddle cycle. J. Differ. Equ. 176, 114–157 (2001)
Dumortier, F., Li, C.: Perturbations from an elliptic Hamiltonian of degree four figure eight-loop. J. Differ. Equ. 188, 512–554 (2003)
Wang, J., Xiao, D.: On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent saddle. J. Differ. Equ. 250, 2227–2243 (2011)
Qi, M., Zhao, L.: Bifurcations of limit cycles from a quintic Hamiltonian system with a figure double-fish. Int. J. Bifurcat. Chaos Appl. Sci. Eng. 23, 1350116 (2013)
Asheghi, R., Zangeneh, H.R.Z.: Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop. Nonlinear Anal. 68, 2957–2976 (2008)
Asheghi, R., Zangeneh, H.R.Z.: Bifurcations of limit cycles from quintic Hamiltonian systems with an eye-figure loop (II). Nonlinear Anal. 69, 4143–4162 (2008)
Asheghi, R., Zangeneh, H.R.Z.: Bifurcations of limit cycles from quintic Hamiltonian system with a double cuspidal loop. Comput. Math. Appl. 59, 1409–1418 (2010)
Zhao, L.: The perturbations of a class of hyper-elliptic Hamilton systems with a double homoclinic loop through a nilpotent saddle. Nonlinear Anal. 95, 374–387 (2014)
Kazemi, R., Zangeneh, H.R.Z., Atabaigi, A.: On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems. Nonlinear Anal. 75, 574–587 (2012)
Zhao, L.: Zeros of Abelian integral for a class of quintic planar vector fields with a double homoclinic loop under non-symmetry perturbations. Acta Math. SINICA 47, 227–240 (2017)
Li, J., Li, C.: Distribution of limit cycles for planar cubic Hamiltonian systems. Acta Math. SINICA 28, 509–521 (1985)
Hong, X., Tan, B.: Bifurcation of limit cycles of a perturbed integrable non-Hamiltonian system. In: Proceedings of the 4th International Workshop on Chaos-Fractal Theories and Applications, pp. 22–26 (2011)
Hong, X.C.: Limit cycles analysis of a perturbed integal Abel non-Hamiltonian system. Ann. Differ. Equ. 3, 263–268 (2012)
Acknowledgment
This work was financially supported by the Natural Science Foundation of China (Grant No. 11761075).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Wang, Y., Hong, L., Hong, X. (2020). Limit Cycles Analysis in a Fifth-Order Vector Field with Asymmetric Perturbation Terms. In: Liu, Y., Wang, L., Zhao, L., Yu, Z. (eds) Advances in Natural Computation, Fuzzy Systems and Knowledge Discovery. ICNC-FSKD 2019. Advances in Intelligent Systems and Computing, vol 1074. Springer, Cham. https://doi.org/10.1007/978-3-030-32456-8_56
Download citation
DOI: https://doi.org/10.1007/978-3-030-32456-8_56
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-32455-1
Online ISBN: 978-3-030-32456-8
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)