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Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips

Science china mathematics 2005年第2期48卷 248-260页

作者： shui shuliang & zhu deming college of mathematics and physics, zhejiang normal university, jinhua 3.1.04, china Department of mathematics, East china normal university, Shanghai 200062, china 1. college of mathematics and physics Zhejiang Normal University 321004 Jinhua China 2. Department of mathematics East China Normal University 200062 Shanghai China

Homoclinic bifurcations in four-dimensional vector fields are investigated by setting up a local coordinate near a homoclinic orbit. This homoclinic orbit is principal but its stable and unstable foliations take inclination flip. The existence, nonexistence, and uniqueness of the 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold 1-periodic orbit and three-fold 1 -periodic orbit are also obtained. It is indicated that the number of periodic orbits bifurcated from this kind of homoclinic orbits depends heavily on the strength of the inclination flip.

关键词： bifurcation, homoclinic orbit, non-resonance, inclination flip, 1-periodic orbit.

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CODIMENSION 3 BIFURCATIONS OFHOMOCLINIC ORBITS WITH ORBITFLIPS AND INCLINATION FLIPS

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Chinese Annals of mathematics,Series B 2004年第4期25卷 555-566页

作者： shuishuliang zhudeming collegeofmathematicsandphysics ZhejiangNormalUniversityJinhua321004ZhejiangChina. Departmentofmathematics EastChinaNormalUniversityShanghai200062China.

The homoclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near the homoclinic orbit. This homoclinic orbit is nonprincipal in the meanings that its positive semi-orbit takes orbit flip and its unstable foliation takes inclination flip. The existence, nonexistence, uniqueness and coexistence of the 1-homoclinic orbit and the 1-periodic orbit are studied. The existence of the twofold periodic orbit and three-fold periodic orbit are also obtained.

关键词： Bifurcation Homoclinic orbit Orbit flip Inclination flip Periodic orbit

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Codimension 3 Non-resonant Bifurcations of Rough Heteroclinic Loops with One Orbit Flip

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Chinese Annals of mathematics,Series B 2006年第6期27卷 657-674页

作者： shuliang shui deming zhu college of mathematics Physics and Information Engineering Zhejiaag Normal University Jinhua 321004 Zhejiang China college of Sciences Shanghai University Shanghai 200436 China Department of mathematics East China Normal University Shanghai 200062 China

Heteroclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near a rough heteroclinic loop. This heteroclinic loop has a principal heteroclinic orbit and a non-principal heteroclinic orbit that takes orbit flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit and l-periodic orbit are studied. The existence of the two-fold or three-fold 1-periodic orbit is also obtained.

关键词： Bifurcation Heteroclinic loop Non-resonance Orbit flip Periodic orbit

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