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.
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
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 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
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.
In this paper, we prove that a planar quadratic systems with a 3rd-order weak focus has at most one limit cycle, and a planar quadratic system with a 2nd-order weak focus has at most two limit cycles.
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.
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.
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 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>
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Supported by the National Natural Science Foundation of China (19671071).
文摘In this paper, we prove that a planar quadratic systems with a 3rd-order weak focus has at most one limit cycle, and a planar quadratic system with a 2nd-order weak focus has at most two limit cycles.
