In this paper, we discuss the limit cycles of the systemdx/dt=y·[1+(A(x)]oy/dt=(-x+δy+α_1x^2+α_2xy+α_5x^2y)[1+B(x)] (1)where A(x)=sum form i=1 to n(a_ix~), B(x)=sum form j=1 to m(β_jx^j) and 1+B(x)>0. We ...In this paper, we discuss the limit cycles of the systemdx/dt=y·[1+(A(x)]oy/dt=(-x+δy+α_1x^2+α_2xy+α_5x^2y)[1+B(x)] (1)where A(x)=sum form i=1 to n(a_ix~), B(x)=sum form j=1 to m(β_jx^j) and 1+B(x)>0. We prove that (1) possesses at most one limit cycle and give out the necessary and sufficient conditions of existence and uniqueness of limit cycles.展开更多
In this paper, we discuss the boundedness of the solutions, the existence andthe uniqueness of the limit cycle of the following cubic differential system:x’=y, y’=-x+δy+a<sub>2</sub>xy+a<sub>4<...In this paper, we discuss the boundedness of the solutions, the existence andthe uniqueness of the limit cycle of the following cubic differential system:x’=y, y’=-x+δy+a<sub>2</sub>xy+a<sub>4</sub>x+a<sub>5</sub>x<sup>2</sup>y. （*）We obtain the following results:（1） System （*） is bounded if and only if （i） a<sub>5</sub>【0, a<sub>4</sub>=0; or （ii） a<sub>5</sub>=0, a<sub>4</sub>【0, δ≤0,-（-8a<sub>4</sub>）<sup>1/2</sup>【a<sub>2</sub>【（-8a<sub>4</sub>）<sup>1/2</sup>.（2） System （*） has no limit cycle if a<sub>5</sub>δ≥0.（3） System （*） has one and only one limit cycle if a<sub>5</sub>δ【0, for a<sub>4</sub>≤0.展开更多
In this paper we consider the existence, uniqueness and nonexistence of limit cycles for the class of planar cubic system x=-y+δx+a2xy+a3x2+a7x3, y=x, where a7≠0.
In this paper, by using the qualitative method, we study a class of Kolmogorov 's ecological system with prey having constant adding rate, discuss the relative position and the character of the equilibriums, the g...In this paper, by using the qualitative method, we study a class of Kolmogorov 's ecological system with prey having constant adding rate, discuss the relative position and the character of the equilibriums, the global stability of the practical equilibriums and give a group of conditions for the boundedness of the solutions, the nonexistence, the existence and the uniqueness of the limit cycle of the system. Most results obtained in papers [1] and [2] are included or generalized.展开更多
Consider a class of Ivlev＇s type predator-prey dynamic systems with prey and predator both having linear density restricts. By using the qualitative methods of ODE, the global stability of positive equilibrium and ex...Consider a class of Ivlev＇s type predator-prey dynamic systems with prey and predator both having linear density restricts. By using the qualitative methods of ODE, the global stability of positive equilibrium and existence and uniqueness of non-small amplitude stable limit cycle are obtained. Especially under certain conditions, it shows that existence and uniqueness of non-small amplitude stable limit cycle is equivalent to the local un-stability of positive equilibrium and the local stability of positive equilibrium implies its global stability. That is to say, the global dynamic of the system is entirely determined by the local stability of the positive equilibrium.展开更多
文摘In this paper, we discuss the limit cycles of the systemdx/dt=y·[1+(A(x)]oy/dt=(-x+δy+α_1x^2+α_2xy+α_5x^2y)[1+B(x)] (1)where A(x)=sum form i=1 to n(a_ix~), B(x)=sum form j=1 to m(β_jx^j) and 1+B(x)>0. We prove that (1) possesses at most one limit cycle and give out the necessary and sufficient conditions of existence and uniqueness of limit cycles.
基金This paper is supported by the China Youth Natural Science Foundation.
文摘In this paper, we discuss the boundedness of the solutions, the existence andthe uniqueness of the limit cycle of the following cubic differential system:x’=y, y’=-x+δy+a<sub>2</sub>xy+a<sub>4</sub>x+a<sub>5</sub>x<sup>2</sup>y. （*）We obtain the following results:（1） System （*） is bounded if and only if （i） a<sub>5</sub>【0, a<sub>4</sub>=0; or （ii） a<sub>5</sub>=0, a<sub>4</sub>【0, δ≤0,-（-8a<sub>4</sub>）<sup>1/2</sup>【a<sub>2</sub>【（-8a<sub>4</sub>）<sup>1/2</sup>.（2） System （*） has no limit cycle if a<sub>5</sub>δ≥0.（3） System （*） has one and only one limit cycle if a<sub>5</sub>δ【0, for a<sub>4</sub>≤0.
文摘In this paper we consider the existence, uniqueness and nonexistence of limit cycles for the class of planar cubic system x=-y+δx+a2xy+a3x2+a7x3, y=x, where a7≠0.
基金This paper was financially supported by the Chinese National Youth Natural Science Funds.
文摘In this paper, by using the qualitative method, we study a class of Kolmogorov 's ecological system with prey having constant adding rate, discuss the relative position and the character of the equilibriums, the global stability of the practical equilibriums and give a group of conditions for the boundedness of the solutions, the nonexistence, the existence and the uniqueness of the limit cycle of the system. Most results obtained in papers [1] and [2] are included or generalized.
文摘Consider a class of Ivlev＇s type predator-prey dynamic systems with prey and predator both having linear density restricts. By using the qualitative methods of ODE, the global stability of positive equilibrium and existence and uniqueness of non-small amplitude stable limit cycle are obtained. Especially under certain conditions, it shows that existence and uniqueness of non-small amplitude stable limit cycle is equivalent to the local un-stability of positive equilibrium and the local stability of positive equilibrium implies its global stability. That is to say, the global dynamic of the system is entirely determined by the local stability of the positive equilibrium.
