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homoclinic Bifurcation of orbit Flip with Resonant Principal Eigenvalues

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Acta Mathematica Sinica,English Series 2006年第3期22卷 855-864页

作者： Tian Si ZHANG De Ming ZHU Department of Mathematics East China Normal University

Codimension-3 bifurcations of an orbit-flip homoclinic orbit with resonant principal eigenvalues are studied for a four-dimensional system. The existence, number, co-existence and noncoexistence of 1-homoclinic orbit, 1-periodic orbit, 2n-homoclinic orbit and 2n-periodic orbit are obtained. The bifurcation surfaces and existence regions are also given.

关键词： orbit flip homoclinic orbit Periodic orbit Resonance Principal eigenvalues

homoclinic orbits in three-dimensional Shilnikov-type chaotic systems

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Chinese Physics B 2013年第9期22卷 312-323页

作者：冯晶晶张琪昌王炜郝淑英 Tianjin Key Labortory of Nonlinear Dynamics and Chaos Control School of Mechanical EngineeringTianjin University State Key Laboratory of Engines Tianjin University School of Mechanical Engineering Tianjin University of Technology

In this paper, the Pad6 approximant and analytic solution in the neighborhood of the initial value are introduced into the process of constructing the Shilnikov type homoclinic trajectories in three-dimensional nonlinear dynamical systems. The P1D controller system with quadratic and cubic nonlinearities, the simplified solar-wind-driven-magnetosphere-ionosphere system, and the human DNA sequence system are considered. With the aid of presenting a new condition, the solutions of solving the boundary-value problems which are formulated for the trajectory and evaluating the initial amplitude values become available. At the same time, the value of the bifurcation parameter is obtained directly, which is almost consistent with the numerical result.

关键词： chaos Shilnikov theorem homoclinic orbit Pad6 approximation

homoclinic orbitS OF THE DAVEY- STEWARTSON EQUATIONS

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Applied Mathematics and Mechanics(English Edition) 2005年第2期26卷 139-141页

作者：张隽郭柏灵沈守枫 Department of Mathematics Zhejiang University of Technology Institute of Applied Physics and Computational Mathematics Beijing 100088 P.R.China Department of Mathematics Zhejiang University Hangzhou 310027 P.R.China

The homoclinic solutions problem of the Davey-Stewartson ( DS) Equations were studied. By using the Hirota's bilinear method, the homoclinic orbits of the Davey-Stewartson Equations were obtained through the dependent... 详细信息

关键词： Davey-Stewartson equation homoclinic orbit Hirota method

homoclinic　orbitS　IN　PERTURBED　GENERALIZED　HAMILTONIAN　SYSTEMS

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Acta Mathematica Scientia 1996年第4期16卷 361-374页

作者：赵晓华李继彬黄克累 Dept. of Math. of Yunnan University Kunming 650091 China Institute of Appl. Math. of Yunnan Province Kunming 650091 China Beijing University of Aero. and Astro. Beijing 100083 China

It this paper we obtain existence and bifurcation theorems for homoclinic orbits in three-dimeensional,time dependent and independent,perturbations of generalized Hamiltonian differential equations defined on three-dimensional Poisson *** we apply them to a truncated spectral model of the quasi-geostrophic flow on a cyclic β-plane.

关键词： bifurcation Poisson bracket Generalized Hamiltonian system homoclinic orbit Melnikov method perturbation theory

homoclinic orbitS FOR SECOND ORDER HAMILTONIAN SYSTEM WITH QUADRATIC GROWTH

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Applied Mathematics(A Journal of Chinese Universities) 1995年第4期10卷 399-410页

作者： WUSHAOPING LIUJIAQUAN DepartmentofAppliedMathematics ZhejiangUniversityHangzhou310027 DepartmentofMathematics PekingUniversityBeijing100871

Some existence and multiplicity of homoclinic orbit for second order Hamiltonian system x-a(t)x + Wx(t, x)=0 are given by means of variational methods, where the potential V(t, x)=-a(t)|s|2 + W(t, s) is quadratic in s at infinity and subquadratic in s at zero, and the function a(t) satisfies the growth condition lim→∞∫_t￣(t+l) a(t)dt=+∞,l∈R￣1.

关键词： Hamitonian System homoclinic orbit variational method quadratic growth.

Limit cycles and homoclinic orbits and their bifurcation of Bogdanov-Takens system

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Applied Mathematics and Mechanics(English Edition) 2008年第9期29卷 1195-1201页

作者：黄赪彪刘佳 Department of Applied Mechanics and Engineering Sun Yat-sen University

A quantitative analysis of limit cycles and homoclinic orbits, and the bifurcation curve for the Bogdanov-Takens system are discussed. The parameter incremental method for approximate analytical-expressions of these problems is given. These analytical-expressions of the limit cycle and homoclinic orbit are shown as the generalized harmonic functions by employing a time transformation. Curves of the parameters and the stability characteristic exponent of the limit cycle versus amplitude are drawn. Some of the limit cycles and homoclinic orbits phase portraits are plotted. The relationship curves of parameters μ and A with amplitude a and the bifurcation diagrams about the parameter are also given. The numerical accuracy of the calculation results is good.

关键词： Bogdanov-Takens system limit cycle homoclinic orbit bifurcation dia-grams analytical-expressions parameter incremental method

Bifurcation of homoclinic orbits with Saddle-Center Equilibrium

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Chinese Annals of Mathematics,Series B 2007年第1期28卷 81-92页

作者： Xingbo LIU Xianlong FU Deming ZHU Department of Mathematics East China Normal University Shanghai 200062 China.

In this paper, the authors develop new global perturbation techniques for detecting the persistence of transversal homoclinic orbits in a more general nondegenerated system with action-angle variable. The unperturbed system is assumed to have saddlecenter type equilibrium whose stable and unstable manifolds intersect in one dimensional manifold, and does not have to be completely integrable or near-integrable. By constructing local coordinate systems near the unperturbed homoclinic orbit, the conditions of existence of transversal homoclinic orbit are obtained, and the existence of periodic orbits bifurcated from homoclinic orbit is also considered.

关键词： Local coordinate system homoclinic orbit Bifurcation

THE EXISTENCE OF homoclinic orbitS FOR HAMILTONIAN SYSTEMS WITH THE POTENTIALS CHANGING SIGN

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Chinese Annals of Mathematics,Series B 1996年第4期17卷 403-410页

作者：费贵华南京大学数学系江苏南京

This paper studies the existence of nontrival homoclinic orbits of the Hamiltonian systems -L(t)q+V′(t,q)=0 by using the critical point theory, where the potential V(t,q)=b(t)W(q) can change sign. Under a new kind of... 详细信息

关键词： homoclinic orbit Critical point theory Hamiltonian system

ON THE homoclinic orbitS IN A CLASS OF TWO-DEGREE-OF-FREEDOM SYSTEMS UNDER THE RESONANCE CONDITIONS

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Applied Mathematics and Mechanics(English Edition) 2001年第3期22卷 340-352页

作者： WANG Mao-nan(王茂南) XU Zhen-yuan(徐振源) Mathematics and Physics Institute Wuxi University of Light Industry Wuxi 214036 P R China Mathematics and Physics Institute Wuxi University of Light Industry Wuxi 214036 P R China

A class of two-degree-of-freedom systems in resonance with an external, parametric excitation is investigated, the existence of the periodic solutions locked to Omega is proved by the use of the method of multiple scales. This systems can be transformed into the systems of Wiggins under some conditions. A calculating formula which determines the existence of homoclinic orbits of the systems is given.

关键词： a two-degree-of-freedom system method of multiple scales periodic solution homoclinic orbit chaos