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Problems in Homoclinic Bifurcation with Higher Dimensions

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Acta Mathematica Sinica,English Series 1998年第3期14卷 341-352页

作者： zhu deming department of mathematics, east china normal university. shanghai 200062. china department of mathematics East China Normal University 200062 Shanghai China

In this paper, a suitable local coordinate system is constructed by using exponential dichotomies and generalizing the Floquet method from periodic systems to nonperiodic systems. Then the Poincare map is established to solve various problems in homoclinic bifurcations with codimension one or two. Bifurcation diagrams and bifurcation curves are given.

关键词： Exponential dichotomy Local coordinates Homoclinic bifurcation Periodic orbit

Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips

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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 321004, china department of mathematics, east china normal university, shanghai 200062, china College of mathematics and Physics Zhejiang Normal University Jinhua China department of mathematics East China Normal University 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.

HOMOCLINIC BIFURCATION WITH CODIMENSION 3

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Chinese Annals of mathematics,Series B 1994年第2期15卷 205-216页

作者： zhu deming(department of Matehematics,east china normal university, shanghai 200062, china) 华东师范大学数学系上海 200062

First it is proved that both the integral of the divergence and the Melnikov function are invariants of the C2 transformation. Then, the problem of the planar homoclinic bifurcation with codimension 3 is considered. It is proved that, in a small neighborhood of the origin in the parameter space of a Cr (r≥5) system, there exist exactly two Cr-1 semi- stable- limit- cycle branching surfaces, and their common boundary is a unique Cr-1 three-multiple- limit-cycle branching curve. The bifurcation pictures and the asymptotic expansions of the bifurcation functions are given. The stability criterion for the homoclinic loop is also obtained when the integral of the divergence is zero. The proof of the auxiliary theorems will be presented in [16].

关键词： Homoclinic bifurcation Codimension Semi-stable-limit-cycle branch Three- multiple-limit- cycle branch.

On Indecomposable Definite Unimodular Hermitian Forms

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Acta Mathematica Sinica,English Series 1994年第2期10卷 113-120页

作者： zhu Fuzu department of mathematics east china normal university shanghai, 200062 china department of mathematics East China Normal University Shanghai China

For any natural numbers m and n≥17 we can construct explicitly indecomposable definite unimodular normal Hermitian lattices of rank n over the ring of algebraic integers Rm in an imaginary quadratic field (-m1/2). It is proved that for any n （in case m=11, there is one exception n=3） there exist indecomposable definite unimodular normal Hermitian R15(R11- lattices of rank n, and we exhibit representatives for each class. In the exceptional case there are no lattices with the desired properties. The method given in this paper can solve completely the problem of constructing indecomposable definite unimodular normal Hermitian Rm-lattices of any rank n for each m.

关键词： show On Indecomposable Definite Unimodular Hermitian Forms Math

CONSTRUCTION OF INDECOMPOSABLEDEFINITE HERMITIAN FORMS

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Chinese Annals of mathematics,Series B 1994年第3期15卷 349-360页

作者： zhu FUZU (department ofmathematics, east china normal university shanghai 200062,china) 华东师范大学数学系上海 200062

This Paper gives a method to construct indecomposable positive definite integral Hermitianforman imnginary quadratic field Q with gin discriminant and g。n *** is shown that for ally natural numbers n and a, there are n-ary Indecomposable positivedefinite intopal Herlllltian lattices over Q(resp. Q)with discriminant a, exceptfor four(resp. one) exceptions. In these exceptional cases there are no lattices with the desiredproperties.

关键词： Indecomposable lattice(form) Minimum of a lattice(form) Minimal、ctor Irreduclble vector.

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

On Non-Decomposable Hermitian Forms over Gaussian Domain

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Acta Mathematica Sinica,English Series 1998年第4期14卷 447-456页

作者： zhu Fuzu, department of mathematics, east china normal university, shanghai 200062, china east china normal university Department of Mathematics Shanghai China (GRID:grid.22069.3f) (ISNI:***)

Methods are presented for the construction of non-decomposable Hermitian forms over the ring of integers of an imaginary quadratic field.

关键词： Non-decomposable form Indecomposable form Odd form Even form Reduced form

Bifurcations of Rough Heteroclinic Loops with Three Saddle Points

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Acta Mathematica Sinica,English Series 2002年第1期18卷 199-208页

作者： JIN Yin Lai zhu De Ming department of mathematics. Linyi Teachers university. Shandong 276005. P. R. china department of mathematics. east china normal university. shanghai 200062. P. R. china department of mathematics. east china normal university. shanghai 200062. P. R. china department of mathematics Linyi Teachers' University Shandong P. R. China department of mathematics East China Normal University Shanghai P. R. China

In this paper, we study the bifurcation problems of rough heteroclinic loups cormecting threc saddle points for a higher-dimensional system. Under some transversal conditions and the nontwisted condition. the existence. uniqueness. nd incoexistencc of thc l-heteroclinic loop with threc or two saddle pomts. l-homoclinic orbit and l-periodic orbit near T are obtained. Nleanwhile, the bifurcation surfaces and existence regions are also given. Moreover. the above bifurcation results are extended to the case for heteroclinic loop with l saddle points.

关键词： Local coordinates Heteroclinic loop Homoclinic orbit Periodic orbit Bifurcation surface

DEGENERATED HOMOCLINIC BIFURCATIONS WITH HIGHER DIMENSIONS

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Chinese Annals of mathematics,Series B 2000年第2期21卷 201-210页

作者： JINYINLAI zhuDEMINC departmentofmathematics EastChinaNormalUniversityShanghai200062China. departmentofmathematics LinyiTeacherCollegeLinyi276005ShandongChina.

The degenerated homoclinic bifurcation for high dimensional system is considered. The existence, uniqueness, and incoexistence of the 1-homclinic orbit and 1-periodic orbit near　Г are studied under the nonresonant c... 详细信息

关键词： Local coordinates Poincare map Homoclinic orbit Periodic orbit 2-fold periodic orbit Resonant condition