Affiliation:
1. Science College, Linyi University, Linyi, Shandong 276005, China
2. School of Mathematics Science, Shandong Normal University, Jinan 250014, China
Abstract
By using the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic orbit as the local current coordinates system of the system in the small neighborhood of the homoclinic orbit, we discuss the bifurcation problems of nondegenerated homoclinic loops. Under the nonresonant condition, existence, uniqueness, and incoexistence of 1-homoclinic loop and 1-periodic orbit, the inexistence ofk-homoclinic loop andk-periodic orbit is obtained. Under the resonant condition, we study the existence of 1-homoclinic loop, 1-periodic orbit, 2-fold 1-periodic orbit, and two 1-periodic orbits; the coexistence of 1-homoclinic loop and 1-periodic orbit. Moreover, we give the corresponding existence fields and bifurcation surfaces. At last, we study the stability of the homoclinic loop for the two cases of non-resonant and resonant, and we obtain the corresponding criterions.
Funder
National Natural Science Foundation of China
Subject
Applied Mathematics,General Immunology and Microbiology,General Biochemistry, Genetics and Molecular Biology,Modeling and Simulation,General Medicine
Cited by
7 articles.
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