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
In this paper, we study the weak type heterodimensional cycle with orbit-flip in its non-transversal orbit by using the local moving frame approach. For the first two subcases, we present the sufficient conditions for...In this paper, we study the weak type heterodimensional cycle with orbit-flip in its non-transversal orbit by using the local moving frame approach. For the first two subcases, we present the sufficient conditions for the existence, uniqueness and non-coexistence of the homoclinic orbit, heteroclinic orbit and periodic orbit. Based on the bifurcation analysis, the bifurcation surfaces and the existence regions are located. And for the third subcase, we theoretically established both the coexistence condition for the homoclinic loop and the periodic orbit and the coexistence condition for the persistent heterodimensional cycle and multiple periodic orbits due to the weak type of the transversal heteroclinic orbit. Moreover, an analytical example is presented for this subcase.展开更多
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-...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.展开更多
A high-codimension homoclinic bifurcation is considered with one orbit flip and two inclination flips accompanied by resonant principal eigenvalues. A local active coordinate system in a small neighborhood of homoclin...A high-codimension homoclinic bifurcation is considered with one orbit flip and two inclination flips accompanied by resonant principal eigenvalues. A local active coordinate system in a small neighborhood of homoclinic orbit is introduced. By analysis of the bifurcation equation, the authors obtain the conditions when the original flip homoclinic orbit is kept or broken. The existence and the existence regions of several double periodic orbits and one triple periodic orbit bifurcations are proved. Moreover, the complicated homoclinic-doubling bifurcations are found and expressed approximately.展开更多
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supported by the Foundation of Zhejiang Sci-Tech University (ZSTU)(Grant No. 11432732611046)National Natural Science Foundation of China (Grant No. 10671069)
文摘In this paper, we study the weak type heterodimensional cycle with orbit-flip in its non-transversal orbit by using the local moving frame approach. For the first two subcases, we present the sufficient conditions for the existence, uniqueness and non-coexistence of the homoclinic orbit, heteroclinic orbit and periodic orbit. Based on the bifurcation analysis, the bifurcation surfaces and the existence regions are located. And for the third subcase, we theoretically established both the coexistence condition for the homoclinic loop and the periodic orbit and the coexistence condition for the persistent heterodimensional cycle and multiple periodic orbits due to the weak type of the transversal heteroclinic orbit. Moreover, an analytical example is presented for this subcase.
文摘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.
基金supported by the National Natural Science Foundation of China(No.11126097)
文摘A high-codimension homoclinic bifurcation is considered with one orbit flip and two inclination flips accompanied by resonant principal eigenvalues. A local active coordinate system in a small neighborhood of homoclinic orbit is introduced. By analysis of the bifurcation equation, the authors obtain the conditions when the original flip homoclinic orbit is kept or broken. The existence and the existence regions of several double periodic orbits and one triple periodic orbit bifurcations are proved. Moreover, the complicated homoclinic-doubling bifurcations are found and expressed approximately.
