Bifurcations of limit cycles from quintic Hamiltonian systems with a double figure eight loop

Authors:
Zang Hong;Zhang Tonghua;Wencheng Chen
Affiliations:
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China and Department of Applied Mathematics, Shandong University of Science and Technology, Tai'an, Shandong 271019, Chin ...;Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China;Department of Applied Mathematics, Shandong University of Science and Technology, Tai'an, Shandong 271019, China
Venue:
Journal of Complexity
Year:
2004

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Abstract

This paper deals with Liénard equations of the form x˙ = y,y˙ = P(x) + yQ(x,y), with P and Q polynomial of degree 5 and 4, respectively. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree six, exhibiting a double figure eight-loop. It is proved that the Hopf cyclicity is two, and it is also given by the new configurations of the limit cycles bifurcated from the homoclinic loop or heteroclinic loop for quintic system with quintic perturbations by using the methods of bifurcation theory and qualitative analysis.