In this paper,we use the canonical forms of homogeneous polynomials of degree 3 to study the global properties of cubic systems =x+P<sub>3</sub>（x,y）,=y+Q<sub>3</sub>（x,y）（0.1） where P<...In this paper,we use the canonical forms of homogeneous polynomials of degree 3 to study the global properties of cubic systems =x+P<sub>3</sub>（x,y）,=y+Q<sub>3</sub>（x,y）（0.1） where P<sub>3</sub> and Q<sub>3</sub> are homogeneous polynomials of degree 3 in x,y.Through this work,we draw an overall outline of such systems.展开更多
This article analyses a non-Lienard type planar cubic system, and a complete qualitative analysis is given for the system, especially the conclusions for the non-existence, existence and uniqueness of limit cycles are...This article analyses a non-Lienard type planar cubic system, and a complete qualitative analysis is given for the system, especially the conclusions for the non-existence, existence and uniqueness of limit cycles are obtained.展开更多
First, we show that the theorem by Hirsch which guarantees the existence of carrying simplex for competitive system on any n-rectangle： {x ∈ R^n ： 0 ≤ xi ≤ ki, i = 1,..., n} still holds. Next, based on the theore...First, we show that the theorem by Hirsch which guarantees the existence of carrying simplex for competitive system on any n-rectangle： {x ∈ R^n ： 0 ≤ xi ≤ ki, i = 1,..., n} still holds. Next, based on the theorem a competitive system with the linear structure saturation defined on the n-rectangle is investigated, which admits a unique （n - 1）- dimensional carrying simplex as a global attractor. Further, we focus on the whole dynamical behavior of the three-dimensional case, which has a unique locally asymptotically stable positive equilibrium. Hopf bifurcations do not occur. We prove that any limit set is either this positive equilibrium or a limit cycle. If limit cycles exist, the number of them is finite. We also give a criterion for the positive equilibrium to be globally asymptotically stable.展开更多
基金Supported by the National Natural Science Foundation of China,No.19371069
文摘In this paper,we use the canonical forms of homogeneous polynomials of degree 3 to study the global properties of cubic systems =x+P<sub>3</sub>（x,y）,=y+Q<sub>3</sub>（x,y）（0.1） where P<sub>3</sub> and Q<sub>3</sub> are homogeneous polynomials of degree 3 in x,y.Through this work,we draw an overall outline of such systems.
文摘This article analyses a non-Lienard type planar cubic system, and a complete qualitative analysis is given for the system, especially the conclusions for the non-existence, existence and uniqueness of limit cycles are obtained.
文摘First, we show that the theorem by Hirsch which guarantees the existence of carrying simplex for competitive system on any n-rectangle： {x ∈ R^n ： 0 ≤ xi ≤ ki, i = 1,..., n} still holds. Next, based on the theorem a competitive system with the linear structure saturation defined on the n-rectangle is investigated, which admits a unique （n - 1）- dimensional carrying simplex as a global attractor. Further, we focus on the whole dynamical behavior of the three-dimensional case, which has a unique locally asymptotically stable positive equilibrium. Hopf bifurcations do not occur. We prove that any limit set is either this positive equilibrium or a limit cycle. If limit cycles exist, the number of them is finite. We also give a criterion for the positive equilibrium to be globally asymptotically stable.
