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 ...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.展开更多
Let L be a double homoclinic loop of a Hamiltonian system on the plane. We obtain a condition under which L generates at most two large limit cycles by perturbations. We also give conditions for the existence of at mo...Let L be a double homoclinic loop of a Hamiltonian system on the plane. We obtain a condition under which L generates at most two large limit cycles by perturbations. We also give conditions for the existence of at most five or six limit cycles which appear near L under perturbations.展开更多
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 existenc...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.展开更多
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 incli...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.展开更多
The bifurcation problems of rough 2-point-loop are studied for the caseρ 1 1 >λ 1 1 ,ρ 2 1 <λ 2 1 ,ρ 1 1 ρ 2 1 <λ 1 1 λ 2 1 , where ?ρ i 1 <0 and λ i 1 >0 are the pair of principal eigenvalues...The bifurcation problems of rough 2-point-loop are studied for the caseρ 1 1 >λ 1 1 ,ρ 2 1 <λ 2 1 ,ρ 1 1 ρ 2 1 <λ 1 1 λ 2 1 , where ?ρ i 1 <0 and λ i 1 >0 are the pair of principal eigenvalues of unperturbed system at saddle point pi, i = 1,2. Under the transversal and nontwisted conditions, the authors obtain some results of the existence of one 1-periodic orbit, one 1-periodic and one 1-homoclinic loop, two 1-periodic orbits and one 2-fold 1-periodic orbit. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graph is drawn.展开更多
New conditions for a planar homoclinic loop to have cyclicity two under multiple parameter perturbations have been obtained. As an application it is proved that a homoclinic loop of a nongeneric cubic Hamiltonian has ...New conditions for a planar homoclinic loop to have cyclicity two under multiple parameter perturbations have been obtained. As an application it is proved that a homoclinic loop of a nongeneric cubic Hamiltonian has cyclicity two under arbitrary quadratic perturbations.展开更多
Under a generic assumption, the existence and the uniqueness of the periodic orbit generating from a homoclinic bifurcation are shown, and the dimensions of its stable and unstable manifolds are given. In the case of ...Under a generic assumption, the existence and the uniqueness of the periodic orbit generating from a homoclinic bifurcation are shown, and the dimensions of its stable and unstable manifolds are given. In the case of a 3-dimensional system, our result revises the stability criterion given in [4,5].展开更多
By generalizing the Floquet method from periodic systems to systems with exponential dichotomy, a local coordinate system is established in a neighborhood of the heteroclinic loop \%Γ\% to study the bifurcation probl...By generalizing the Floquet method from periodic systems to systems with exponential dichotomy, a local coordinate system is established in a neighborhood of the heteroclinic loop \%Γ\% to study the bifurcation problems of homoclinic and periodic orbits. Asymptotic expressions of the bifurcation surfaces and their relative positions are given. The results obtained in literature concerned with the 1\|hom bifurcation surfaces are improved and extended to the nontransversal case. Existence regions of the 1\|per orbits bifurcated from Γ are described, and the uniqueness and incoexistence of the 1\|hom and 1\|per orbit and the inexistence of the 2\|hom and 2\|per orbit are also obtained.展开更多
It has been found that some nonlinear wave equations have one-loop soliton solutions. What is the dynamical behavior of the so-called one-loop soliton solution? To answer this question, the travelling wave solutions f...It has been found that some nonlinear wave equations have one-loop soliton solutions. What is the dynamical behavior of the so-called one-loop soliton solution? To answer this question, the travelling wave solutions for four nonlinear wave equations are discussed. Exact explicit parametric representations of some special travelling wave solutions are given. The results of this paper show that a loop solution consists of three different breaking travelling wave solutions. It is not one real loop soliton travelling wave solution.展开更多
文摘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.
文摘Let L be a double homoclinic loop of a Hamiltonian system on the plane. We obtain a condition under which L generates at most two large limit cycles by perturbations. We also give conditions for the existence of at most five or six limit cycles which appear near L under perturbations.
基金Project supported byr the National Natural Science Foundation of China (100710122)Shanghai Municipal Foundation of Selected Academic Research.
文摘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.
基金This paper was completed when the first author was visiting East China Normal University.This work was supported by the National Natural Science Foundation of China(Grant No.10071022).
文摘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.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10071022)the Shanghai Priority Academic Discipline.
文摘The bifurcation problems of rough 2-point-loop are studied for the caseρ 1 1 >λ 1 1 ,ρ 2 1 <λ 2 1 ,ρ 1 1 ρ 2 1 <λ 1 1 λ 2 1 , where ?ρ i 1 <0 and λ i 1 >0 are the pair of principal eigenvalues of unperturbed system at saddle point pi, i = 1,2. Under the transversal and nontwisted conditions, the authors obtain some results of the existence of one 1-periodic orbit, one 1-periodic and one 1-homoclinic loop, two 1-periodic orbits and one 2-fold 1-periodic orbit. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graph is drawn.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 19531070 and 19771037)
文摘New conditions for a planar homoclinic loop to have cyclicity two under multiple parameter perturbations have been obtained. As an application it is proved that a homoclinic loop of a nongeneric cubic Hamiltonian has cyclicity two under arbitrary quadratic perturbations.
基金Supported by the National Natural Science Foundation of China
文摘Under a generic assumption, the existence and the uniqueness of the periodic orbit generating from a homoclinic bifurcation are shown, and the dimensions of its stable and unstable manifolds are given. In the case of a 3-dimensional system, our result revises the stability criterion given in [4,5].
文摘By generalizing the Floquet method from periodic systems to systems with exponential dichotomy, a local coordinate system is established in a neighborhood of the heteroclinic loop \%Γ\% to study the bifurcation problems of homoclinic and periodic orbits. Asymptotic expressions of the bifurcation surfaces and their relative positions are given. The results obtained in literature concerned with the 1\|hom bifurcation surfaces are improved and extended to the nontransversal case. Existence regions of the 1\|per orbits bifurcated from Γ are described, and the uniqueness and incoexistence of the 1\|hom and 1\|per orbit and the inexistence of the 2\|hom and 2\|per orbit are also obtained.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10671179)the Natural Science Foundation of Yunnan Province (Grant No. 2005A0013M)
文摘It has been found that some nonlinear wave equations have one-loop soliton solutions. What is the dynamical behavior of the so-called one-loop soliton solution? To answer this question, the travelling wave solutions for four nonlinear wave equations are discussed. Exact explicit parametric representations of some special travelling wave solutions are given. The results of this paper show that a loop solution consists of three different breaking travelling wave solutions. It is not one real loop soliton travelling wave solution.
