In this paper, the authors investigate the invariant cones of quadratic homoge- neous polynomial vector fields in three variables. Necessary and sufficient conditions for the existence of non-isolated invariant closed...In this paper, the authors investigate the invariant cones of quadratic homoge- neous polynomial vector fields in three variables. Necessary and sufficient conditions for the existence of non-isolated invariant closed cones are obtained by the algebraic expressions in terms of the coefficients of certain quadratic homogeneous polynomials.展开更多
For planar analytic homogcneous vector fields, the existence of periodic orbits and the noncxistence of limit sets arc verilied. It is concluded that spacial analytic homlogencous vector tleld of order in has no limit...For planar analytic homogcneous vector fields, the existence of periodic orbits and the noncxistence of limit sets arc verilied. It is concluded that spacial analytic homlogencous vector tleld of order in has no limit sets for any m>1. Similar results arc extended to highel-dimensional polynomial homogeneous vector fields under certain conditions.展开更多
In this paper, we investigate the invariant cones of cubic homogeneous polynomial vector fields with three variables. Sufficient conditions for the existence of invariant non-isolated closed cones are obtained as alge...In this paper, we investigate the invariant cones of cubic homogeneous polynomial vector fields with three variables. Sufficient conditions for the existence of invariant non-isolated closed cones are obtained as algebraic expressions in terms of the coefficients of certain cubic homogeneous polynomials.展开更多
In this paper, the authors prove that the flows of homogeneous vector field Q(x) at infinityare topologically equivalent to the flows of the tangent vector field QT(u) (u∈S2) on thesphere S2, and show the theorems fo...In this paper, the authors prove that the flows of homogeneous vector field Q(x) at infinityare topologically equivalent to the flows of the tangent vector field QT(u) (u∈S2) on thesphere S2, and show the theorems for the global topological classification of Q(x). They derivethe necessary and sufficient conditions for the global asymptotic stability and the boundednessof vector field Q(x), and obtain the criterion for the global topological equivalence of twohomogeneous vector fields.展开更多
In this paper, .we find a bridge connecting a class of vector fields in R3 with the planar vector fields and give a criterion of the existence of closed orbits, heteroclinic orbits and.homoclinic orbits of a class of ...In this paper, .we find a bridge connecting a class of vector fields in R3 with the planar vector fields and give a criterion of the existence of closed orbits, heteroclinic orbits and.homoclinic orbits of a class of vector fields in R3. All the possible nonwandering sets of this class of vector fields fall into three classes： （i） singularities; （ii） closed orbits; （iii） graphs of unions of singularities and the trajectories connecting them. The necessary and sufficient conditions for the boundedness of the vector fields are also obtained.展开更多
In this paper,we investigate the isolated closed orbits of two types of cubic vector fields in R^3 by using the idea of central projection transformation,which sets up a bridge connecting the vector field X(x)in R^3 w...In this paper,we investigate the isolated closed orbits of two types of cubic vector fields in R^3 by using the idea of central projection transformation,which sets up a bridge connecting the vector field X(x)in R^3 with the planar vector fields.We have proved that the cubic vector field in R^3 can have two isolated closed orbits or one closed orbit on the invariant cone.As an application of this result,we have shown that a class of 3-dimensional cubic system has at least 10 isolated closed orbits located on 5 invariant cones,and another type of 3-dimensional cubic system has at least 26 isolated closed orbits located on 13 invariant cones or 26 invariant cones.展开更多
In this paper, a bridge between near-homogeneous and homogeneous vector fields in R 3 is found. By the relationship between homogeneous vector fields and the induced tangent vector fields of two-dimensional manifold S...In this paper, a bridge between near-homogeneous and homogeneous vector fields in R 3 is found. By the relationship between homogeneous vector fields and the induced tangent vector fields of two-dimensional manifold S 2 , we prove the existence of at least 5 isolated closed orbits for a class of n + 1 (n ≥ 2) systems in R 3 , which are located on the five invariant closed cones of the system.展开更多
文摘In this paper, the authors investigate the invariant cones of quadratic homoge- neous polynomial vector fields in three variables. Necessary and sufficient conditions for the existence of non-isolated invariant closed cones are obtained by the algebraic expressions in terms of the coefficients of certain quadratic homogeneous polynomials.
文摘For planar analytic homogcneous vector fields, the existence of periodic orbits and the noncxistence of limit sets arc verilied. It is concluded that spacial analytic homlogencous vector tleld of order in has no limit sets for any m>1. Similar results arc extended to highel-dimensional polynomial homogeneous vector fields under certain conditions.
文摘In this paper, we investigate the invariant cones of cubic homogeneous polynomial vector fields with three variables. Sufficient conditions for the existence of invariant non-isolated closed cones are obtained as algebraic expressions in terms of the coefficients of certain cubic homogeneous polynomials.
文摘In this paper, the authors prove that the flows of homogeneous vector field Q(x) at infinityare topologically equivalent to the flows of the tangent vector field QT(u) (u∈S2) on thesphere S2, and show the theorems for the global topological classification of Q(x). They derivethe necessary and sufficient conditions for the global asymptotic stability and the boundednessof vector field Q(x), and obtain the criterion for the global topological equivalence of twohomogeneous vector fields.
基金Supported by National Natural Science Foundation of China （Grant No. 10771081）
文摘In this paper, .we find a bridge connecting a class of vector fields in R3 with the planar vector fields and give a criterion of the existence of closed orbits, heteroclinic orbits and.homoclinic orbits of a class of vector fields in R3. All the possible nonwandering sets of this class of vector fields fall into three classes： （i） singularities; （ii） closed orbits; （iii） graphs of unions of singularities and the trajectories connecting them. The necessary and sufficient conditions for the boundedness of the vector fields are also obtained.
基金Supported by the National Nature Science Foundation of China(Grant Nos.11871238,11971405)selfdetermined research funds of CCNU from the collegesbasic research and operation of MOE(Grant No.CCNU16JCZX10)+1 种基金the Natural Science Foundation of Fujian Province of China(Grant No.2015J05016)the Fundamental Research Funds of the South-Central University for Nationalities(Grant No.CZQ13016)。
文摘In this paper,we investigate the isolated closed orbits of two types of cubic vector fields in R^3 by using the idea of central projection transformation,which sets up a bridge connecting the vector field X(x)in R^3 with the planar vector fields.We have proved that the cubic vector field in R^3 can have two isolated closed orbits or one closed orbit on the invariant cone.As an application of this result,we have shown that a class of 3-dimensional cubic system has at least 10 isolated closed orbits located on 5 invariant cones,and another type of 3-dimensional cubic system has at least 26 isolated closed orbits located on 13 invariant cones or 26 invariant cones.
基金supported by the National Natural Science Foundation of China (No.10701037, No.10871080 and No.10771081)
文摘In this paper, a bridge between near-homogeneous and homogeneous vector fields in R 3 is found. By the relationship between homogeneous vector fields and the induced tangent vector fields of two-dimensional manifold S 2 , we prove the existence of at least 5 isolated closed orbits for a class of n + 1 (n ≥ 2) systems in R 3 , which are located on the five invariant closed cones of the system.
