We study a class of quartic polynomial Poincare equations by applying a recurrence formula of focal value. We give the necessary and sufficient conditions for the origin to be a center, and prove that the order of fin...We study a class of quartic polynomial Poincare equations by applying a recurrence formula of focal value. We give the necessary and sufficient conditions for the origin to be a center, and prove that the order of fine focus at the origin for this class of equations is at most 6. Key words quartic polynomial Poincare equation - center - fine focus - order CLC number O 175. 12 Foundation item: Supported by the National Natural Science Foundation of China (19531070)Biography: TIAN De-sheng (1966-), male, Ph. D candidate, research direction: qualitative theory of differential equation.展开更多
In this paper we study the existence and non-existence of limit cycles for aclass of system which has two finite critical points: one of them is a fine focusand the other is a fine saddle. The results obtained are alm...In this paper we study the existence and non-existence of limit cycles for aclass of system which has two finite critical points: one of them is a fine focusand the other is a fine saddle. The results obtained are almost complete.展开更多
Gyllenberg and Yan(Discrete Contin Dyn Syst Ser B 11(2):347–352,2009)presented a system in Zeeman’s class 30 of 3-dimensional Lotka-Volterra(3D LV)competitive systems to admit at least two limit cycles,one of which ...Gyllenberg and Yan(Discrete Contin Dyn Syst Ser B 11(2):347–352,2009)presented a system in Zeeman’s class 30 of 3-dimensional Lotka-Volterra(3D LV)competitive systems to admit at least two limit cycles,one of which is generated by the Hopf bifurcation and the other is obtained by the Poincaré-Bendixson theorem.Yu et al.(J Math Anal Appl 436:521–555,2016,Sect.3.4)recalculated the first Liapunov coefficient of Gyllenberg and Yan’s system to be positive,rather than negative as in Gyllenberg and Yan(2009),and pointed out that the Poincaré-Bendixson theorem is not applicable for that system.Jiang et al.(J Differ Equ 284:183–218,2021,p.213)proposed an open question:“whether Zeeman’s class 30 can be rigorously proved to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem?”This paper provides four systems in Zeeman’s class 30 to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem and gives an answer to the above question.展开更多
A Kukles system with two fine foci is considered. We prove if the two finefoci have the same order, then the highest order of each fine focus is two; if thetwo fine fool have different order, and if the highest order ...A Kukles system with two fine foci is considered. We prove if the two finefoci have the same order, then the highest order of each fine focus is two; if thetwo fine fool have different order, and if the highest order of one of these two finefoci is one, then the highest order of the other is five. Based on these results wecan further prove that a Kukles system with two fine foci can generate at leastsix limit cycles.展开更多
We study the global qualitative properties of the well-known Kukles systems (1) below. Firstly, the number of critical points in case (1) has a center or a fine focus.
In this paper we study the relation of trajectories and limit cycles between index-inverse differential systems, especially, index-inverse quadratic differential systems.
文摘We study a class of quartic polynomial Poincare equations by applying a recurrence formula of focal value. We give the necessary and sufficient conditions for the origin to be a center, and prove that the order of fine focus at the origin for this class of equations is at most 6. Key words quartic polynomial Poincare equation - center - fine focus - order CLC number O 175. 12 Foundation item: Supported by the National Natural Science Foundation of China (19531070)Biography: TIAN De-sheng (1966-), male, Ph. D candidate, research direction: qualitative theory of differential equation.
文摘In this paper we study the existence and non-existence of limit cycles for aclass of system which has two finite critical points: one of them is a fine focusand the other is a fine saddle. The results obtained are almost complete.
基金the National Natural Science Foundation of China(NSFC)under Grant No.12171321.
文摘Gyllenberg and Yan(Discrete Contin Dyn Syst Ser B 11(2):347–352,2009)presented a system in Zeeman’s class 30 of 3-dimensional Lotka-Volterra(3D LV)competitive systems to admit at least two limit cycles,one of which is generated by the Hopf bifurcation and the other is obtained by the Poincaré-Bendixson theorem.Yu et al.(J Math Anal Appl 436:521–555,2016,Sect.3.4)recalculated the first Liapunov coefficient of Gyllenberg and Yan’s system to be positive,rather than negative as in Gyllenberg and Yan(2009),and pointed out that the Poincaré-Bendixson theorem is not applicable for that system.Jiang et al.(J Differ Equ 284:183–218,2021,p.213)proposed an open question:“whether Zeeman’s class 30 can be rigorously proved to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem?”This paper provides four systems in Zeeman’s class 30 to admit at least two limit cycles by the Hopf theorem and the Poincaré-Bendixson theorem and gives an answer to the above question.
文摘A Kukles system with two fine foci is considered. We prove if the two finefoci have the same order, then the highest order of each fine focus is two; if thetwo fine fool have different order, and if the highest order of one of these two finefoci is one, then the highest order of the other is five. Based on these results wecan further prove that a Kukles system with two fine foci can generate at leastsix limit cycles.
基金Project supported by the Natural Science Foundation of China.
文摘We study the global qualitative properties of the well-known Kukles systems (1) below. Firstly, the number of critical points in case (1) has a center or a fine focus.
文摘In this paper we study the relation of trajectories and limit cycles between index-inverse differential systems, especially, index-inverse quadratic differential systems.
