The existence of the Hopf bifurcation of a complex ordinary differential equation system in the complex domain is studied in this paper by using the complex qualitative theory.In the complex domain,we conclude that th...The existence of the Hopf bifurcation of a complex ordinary differential equation system in the complex domain is studied in this paper by using the complex qualitative theory.In the complex domain,we conclude that the Hopf bifurcation appears for both directions of the parameter μ. The formulae of the Hopf bifurcation are also given in this paper.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
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
The authors study the bifurcation problems of rough heteroclinic loop connecting threesaddle points for the case β1 ＞ 1, β2 ＞ 1, β3 ＜ 1 and β1β2β3 ＜ 1. The existence, number, co-existence and incoexistence o...The authors study the bifurcation problems of rough heteroclinic loop connecting threesaddle points for the case β1 ＞ 1, β2 ＞ 1, β3 ＜ 1 and β1β2β3 ＜ 1. The existence, number, co-existence and incoexistence of 2-point-loop, 1-homoclinic orbit and 1-periodic orbit are studied.Meanwhile, the bifurcation surfaces and existence regions are given.展开更多
By using the linear independent fundamental solutions of the linearvariational equation along the heteroclinic loop to establish a suitable local coordinate system insome small tubular neighborhood of the heteroclinic...By using the linear independent fundamental solutions of the linearvariational equation along the heteroclinic loop to establish a suitable local coordinate system insome small tubular neighborhood of the heteroclinic loop, the Poincaré map is constructed to studythe bifurcation problems of a fine 3–point loop in higher dimensional space. Under some transversalconditions and the non–twisted condition, the existence, coexistence and incoexistence of2–point–loop, 1–homoclinic orbit, simple 1–periodic orbit and 2–fold 1–periodic orbit, and thenumber of 1–periodic orbits are studied. Moreover, the bifurcation surfaces and existence regionsare given. Lastly, the above bifurcation results are applied to a planar system and an insidestability criterion is obtained.展开更多
文摘The existence of the Hopf bifurcation of a complex ordinary differential equation system in the complex domain is studied in this paper by using the complex qualitative theory.In the complex domain,we conclude that the Hopf bifurcation appears for both directions of the parameter μ. The formulae of the Hopf bifurcation are also given in this paper.
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 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 (No.10071022) the Shanghai Priority Academic Discipline.
文摘The authors study the bifurcation problems of rough heteroclinic loop connecting threesaddle points for the case β1 ＞ 1, β2 ＞ 1, β3 ＜ 1 and β1β2β3 ＜ 1. The existence, number, co-existence and incoexistence of 2-point-loop, 1-homoclinic orbit and 1-periodic orbit are studied.Meanwhile, the bifurcation surfaces and existence regions are given.
基金This work is supported by the National Natural Science Foundation of China(10371040)the Shanghai Priority Academic Disciplinesthe Scientific Research Foundation of Linyi Teacher's University 37C29,34C23,34C37
文摘By using the linear independent fundamental solutions of the linearvariational equation along the heteroclinic loop to establish a suitable local coordinate system insome small tubular neighborhood of the heteroclinic loop, the Poincaré map is constructed to studythe bifurcation problems of a fine 3–point loop in higher dimensional space. Under some transversalconditions and the non–twisted condition, the existence, coexistence and incoexistence of2–point–loop, 1–homoclinic orbit, simple 1–periodic orbit and 2–fold 1–periodic orbit, and thenumber of 1–periodic orbits are studied. Moreover, the bifurcation surfaces and existence regionsare given. Lastly, the above bifurcation results are applied to a planar system and an insidestability criterion is obtained.
