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
This paper studies a class of quartic system which is more general and realistic than the quartic accompanying system.x＇=-y＋ex＋lx^2＋mxy＋ny^2,y＇=x（1-Ay）（1＋Cy^2）,（＊）where C 〉 0. Sufficient conditions are ...This paper studies a class of quartic system which is more general and realistic than the quartic accompanying system.x＇=-y＋ex＋lx^2＋mxy＋ny^2,y＇=x（1-Ay）（1＋Cy^2）,（＊）where C 〉 0. Sufficient conditions are obtained for the uniqueness of limit cycle of system （＊） and some more in-depth conclusion such as Hopf bifurcation.展开更多
We present a new criterion for studying the uniqueness of limit cycle of a generalized Liénard system (E). It generalizes the traditional criterion concerning above topic.
We transform the quadratic system into the special system of Type (Ⅲ)a=0' and hence a string sufficient conditions are established to ensure that the considered system has at most one limit cycle.
In this paper, we have proved several theorems which guarantee that the Lienard equation has at least one or n limit cycles without using the traditional assumption G Thus some results in [3-5] are extended. The limit...In this paper, we have proved several theorems which guarantee that the Lienard equation has at least one or n limit cycles without using the traditional assumption G Thus some results in [3-5] are extended. The limit cycles can be located by our theorems. Theorems 3 and 4 give sufficient conditions for the existence of n limit cycles having no need of the conditions that the function F(x) is odd or 'nth order compatible with each other' or 'nth order contained in each other'.展开更多
In this paper, we first give a necessary and sufficient condition of a quadratic system with three finite critical points being bounded, and then, we use the methods and conclusions of [11] to provide some uniqueness ...In this paper, we first give a necessary and sufficient condition of a quadratic system with three finite critical points being bounded, and then, we use the methods and conclusions of [11] to provide some uniqueness theorems of limit cycles for bounded quadratic systems. As well, we prove that any bounded quadratic system can not have (2, 2)-distribution of limit cycles according to these uniqueness theorems.展开更多
A class of cubic system, which is an accompany system of a quadratic differential one, is studied. It is proved that the system has at most one limit cycle, and the critical point at infinity is a higher order one. Th...A class of cubic system, which is an accompany system of a quadratic differential one, is studied. It is proved that the system has at most one limit cycle, and the critical point at infinity is a higher order one. The structure and algebraic character of the critical point at infinity are obtained.展开更多
In this paper,we consider a class of quartic system,which is more general and realistic than the quartic accompanying system. Consequently,we obtain sufficient conditions concerning the uniqueness of limit cycle as we...In this paper,we consider a class of quartic system,which is more general and realistic than the quartic accompanying system. Consequently,we obtain sufficient conditions concerning the uniqueness of limit cycle as well as some other in-depth conclusions.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 ref.[1]of § 20,Ye Yanqian has investigated the impossibility of（2,2）distribution of lim-it cycles of quadratic systems,where the footnote 1)on p.553 gives the following conjecture:The quadratic system（Ⅲ）&...IN ref.[1]of § 20,Ye Yanqian has investigated the impossibility of（2,2）distribution of lim-it cycles of quadratic systems,where the footnote 1)on p.553 gives the following conjecture:The quadratic system（Ⅲ）<sub>m=0</sub>展开更多
Without loss of generality, the quadratic system （Ⅱ）<sub>m=0</sub> can be assumed as follows:Generally, system （1） has four singular points, focus （node） 0（0,0）, R（-1/a,y<sub>2</sub>...Without loss of generality, the quadratic system （Ⅱ）<sub>m=0</sub> can be assumed as follows:Generally, system （1） has four singular points, focus （node） 0（0,0）, R（-1/a,y<sub>2</sub>）, saddle N（0, 1）, M（-1/a,y<sub>1</sub>）, where y<sub>1, 2</sub>=[a±（a<sup>2</sup>-4（l-aδ））<sup>1/2</sup>]/2a.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,we consider an accompany system concerning some class of cubic system. We then prove that the system has at most one limit cycle. Finally,we obtain the topological structure of both the critical points a...In this paper,we consider an accompany system concerning some class of cubic system. We then prove that the system has at most one limit cycle. Finally,we obtain the topological structure of both the critical points at infinity and the singular points lying on invariant lines.展开更多
Consider a three-dimensional autonomous system of the form of =f(u), u∈R^3, (0.1) where f(0)=0, and Df(0)=. By transforming eq. (0.1) into a normal form equation and then unfolding the truncated equation, one can obt...Consider a three-dimensional autonomous system of the form of =f(u), u∈R^3, (0.1) where f(0)=0, and Df(0)=. By transforming eq. (0.1) into a normal form equation and then unfolding the truncated equation, one can obtain a plane system of the form展开更多
In this paper, we continue to discuss the uniqueness of limit cycle of the quadratic system by using the quadratic curve without contact and several new criteria for the uniqueness have been obtained.
To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critic...To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation,the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through saddle S1 on 1 ＋ax-y = 0,which has the same stability with the limit cycle created by Hopf bifurcation,then the uniqueness of limit cycles in such cases can be proved. If it is a homoclinic cycle passing through saddle N on x= 0,which has the different stability from the limit cycle created by Hopf bifurcation,then it will be a case of two limit cycles. For the case when the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity,the discussion of the paper shows that the number of limit cycles will change from one to two depending on the different values of parameters of system.展开更多
In this paper, the nonexistence, existence and the number of limit cycles for a class of differential systems with positive definite polynomial are considered, and the results obtained generalize and supplement those ...In this paper, the nonexistence, existence and the number of limit cycles for a class of differential systems with positive definite polynomial are considered, and the results obtained generalize and supplement those of [1].展开更多
In this paper, we study the number of limit cycles appeared in Hopf bifurcations of a Lienard system with multiple parameters. As an application to some polynomial Lienard systems of the form x= y, y= -gin（x） - fn...In this paper, we study the number of limit cycles appeared in Hopf bifurcations of a Lienard system with multiple parameters. As an application to some polynomial Lienard systems of the form x= y, y= -gin（x） - fn（X）y, we obtain a new lower bound of maximal number of limit cycles which appear in Hopf bifurcation for arbitrary degrees m and n.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Supported by the Natural Science Foundation of Fujian Province(Z0511052,2006J0209)the Foundation of Fujian Education Department(JA04158,JA04274)and the Foundation of Developing ScienceTechnology of Fuzhou University(2005-QX-20)
文摘This paper studies a class of quartic system which is more general and realistic than the quartic accompanying system.x＇=-y＋ex＋lx^2＋mxy＋ny^2,y＇=x（1-Ay）（1＋Cy^2）,（＊）where C 〉 0. Sufficient conditions are obtained for the uniqueness of limit cycle of system （＊） and some more in-depth conclusion such as Hopf bifurcation.
文摘In this paper, the uniqueness of limit cycle of a special polynomial Lienard system is discussed and some results under certain conditions are given.
文摘We present a new criterion for studying the uniqueness of limit cycle of a generalized Liénard system (E). It generalizes the traditional criterion concerning above topic.
文摘We transform the quadratic system into the special system of Type (Ⅲ)a=0' and hence a string sufficient conditions are established to ensure that the considered system has at most one limit cycle.
文摘In this paper, we have proved several theorems which guarantee that the Lienard equation has at least one or n limit cycles without using the traditional assumption G Thus some results in [3-5] are extended. The limit cycles can be located by our theorems. Theorems 3 and 4 give sufficient conditions for the existence of n limit cycles having no need of the conditions that the function F(x) is odd or 'nth order compatible with each other' or 'nth order contained in each other'.
文摘In this paper, we first give a necessary and sufficient condition of a quadratic system with three finite critical points being bounded, and then, we use the methods and conclusions of [11] to provide some uniqueness theorems of limit cycles for bounded quadratic systems. As well, we prove that any bounded quadratic system can not have (2, 2)-distribution of limit cycles according to these uniqueness theorems.
基金This work is supported by the National Natural Science Foundation of China (Tian Yuan Foundation) (10426010) Natural Science Foundation of Fujian Province (Z0511052)Fujian Educational Bureau (JA04274).
文摘A class of cubic system, which is an accompany system of a quadratic differential one, is studied. It is proved that the system has at most one limit cycle, and the critical point at infinity is a higher order one. The structure and algebraic character of the critical point at infinity are obtained.
基金This work was supported by the Natural Science Foundation of Fujian (Z0511052)the Foundation of Fujian Education Bureau (JA04158, JA04274)the Foundation of Developing Science and Technology of Fuzhou University (2005-QX-20).
文摘In this paper,we consider a class of quartic system,which is more general and realistic than the quartic accompanying system. Consequently,we obtain sufficient conditions concerning the uniqueness of limit cycle as well as some other in-depth conclusions.
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 ref.[1]of § 20,Ye Yanqian has investigated the impossibility of（2,2）distribution of lim-it cycles of quadratic systems,where the footnote 1)on p.553 gives the following conjecture:The quadratic system（Ⅲ）<sub>m=0</sub>
基金Project supported by the National Natural Science Foundation of China
文摘Without loss of generality, the quadratic system （Ⅱ）<sub>m=0</sub> can be assumed as follows:Generally, system （1） has four singular points, focus （node） 0（0,0）, R（-1/a,y<sub>2</sub>）, saddle N（0, 1）, M（-1/a,y<sub>1</sub>）, where y<sub>1, 2</sub>=[a±（a<sup>2</sup>-4（l-aδ））<sup>1/2</sup>]/2a.
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The National Natural Science Foundation of China (10371006)Natural Science Foundation of Anhui Province (050460103)Anhui Educational Bureau (2005kj031ZD).
文摘In this paper,we consider an accompany system concerning some class of cubic system. We then prove that the system has at most one limit cycle. Finally,we obtain the topological structure of both the critical points at infinity and the singular points lying on invariant lines.
基金the National Natural Science Foundation of China.
文摘Consider a three-dimensional autonomous system of the form of =f(u), u∈R^3, (0.1) where f(0)=0, and Df(0)=. By transforming eq. (0.1) into a normal form equation and then unfolding the truncated equation, one can obtain a plane system of the form
文摘In this paper, we continue to discuss the uniqueness of limit cycle of the quadratic system by using the quadratic curve without contact and several new criteria for the uniqueness have been obtained.
基金Project supported by the National Natural Science Foundation of China （10471066）.
文摘To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation,the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through saddle S1 on 1 ＋ax-y = 0,which has the same stability with the limit cycle created by Hopf bifurcation,then the uniqueness of limit cycles in such cases can be proved. If it is a homoclinic cycle passing through saddle N on x= 0,which has the different stability from the limit cycle created by Hopf bifurcation,then it will be a case of two limit cycles. For the case when the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity,the discussion of the paper shows that the number of limit cycles will change from one to two depending on the different values of parameters of system.
文摘In this paper, the nonexistence, existence and the number of limit cycles for a class of differential systems with positive definite polynomial are considered, and the results obtained generalize and supplement those of [1].
基金supported by National Natural Science Foundation of China (Grant No.11271261)
文摘In this paper, we study the number of limit cycles appeared in Hopf bifurcations of a Lienard system with multiple parameters. As an application to some polynomial Lienard systems of the form x= y, y= -gin（x） - fn（X）y, we obtain a new lower bound of maximal number of limit cycles which appear in Hopf bifurcation for arbitrary degrees m and n.
