A sufficient condition for fcth Lyapunov value to be zero for planar polynomial vector fields is given, which extends the result of "jumping property’ of Lyapunov values obtained by Wang Duo to more general case...A sufficient condition for fcth Lyapunov value to be zero for planar polynomial vector fields is given, which extends the result of "jumping property’ of Lyapunov values obtained by Wang Duo to more general cases. A concrete example that the origin cannot be weak focus of order 1, 2, 4, 5, 8 is presented.展开更多
It is proved that the quadratic system with a weak focus and a strong focus has at most one limit cycle around the strong focus, and as the weak focus is a 2nd order(or 3rd order) weak focus the quadratic system ha...It is proved that the quadratic system with a weak focus and a strong focus has at most one limit cycle around the strong focus, and as the weak focus is a 2nd order(or 3rd order) weak focus the quadratic system has at most two(one) limit cycles which have (1,1) distribution ((0,1) distribution).展开更多
In this paper, we investigate the maximal number of limit cycles surrounding a first order weak focus for the quadratic differential system. And proved such a system has at most two limit cycles under some certain con...In this paper, we investigate the maximal number of limit cycles surrounding a first order weak focus for the quadratic differential system. And proved such a system has at most two limit cycles under some certain conditions.展开更多
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.
For any second order autonomous differential system with a polynomial of degree n on its right-hand side, what the maximum order of fineness of its weak focal point, weak saddle point in real domain and weak critical ...For any second order autonomous differential system with a polynomial of degree n on its right-hand side, what the maximum order of fineness of its weak focal point, weak saddle point in real domain and weak critical singular point in complex domain is (M(n)=?)? This problem relates to Hilbert’s 16th problem closely. Inequality M(3)≥7 have展开更多
In this paper we study the variation of limit cycles around different foci when a coefficient in the equation of the quadratic differential system varies.
For the planar Z2-equivariant cubic systems having two elementary focuses, the characterization of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved. The necessary...For the planar Z2-equivariant cubic systems having two elementary focuses, the characterization of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved. The necessary and sufficient conditions for the existence of the bi-center are obtained. On the basis of this work, in this paper, we show that under small Z2-equivariant cubic perturbations, this cubic system has at least 13 limit cycles with the scheme 1 6 ∪ 6.展开更多
基金National Natural Science Foundation of China,State Education Commission of China and Tsinghua University.
文摘A sufficient condition for fcth Lyapunov value to be zero for planar polynomial vector fields is given, which extends the result of "jumping property’ of Lyapunov values obtained by Wang Duo to more general cases. A concrete example that the origin cannot be weak focus of order 1, 2, 4, 5, 8 is presented.
文摘It is proved that the quadratic system with a weak focus and a strong focus has at most one limit cycle around the strong focus, and as the weak focus is a 2nd order(or 3rd order) weak focus the quadratic system has at most two(one) limit cycles which have (1,1) distribution ((0,1) distribution).
文摘In this paper, we investigate the maximal number of limit cycles surrounding a first order weak focus for the quadratic differential system. And proved such a system has at most two limit cycles under some certain conditions.
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.
文摘For any second order autonomous differential system with a polynomial of degree n on its right-hand side, what the maximum order of fineness of its weak focal point, weak saddle point in real domain and weak critical singular point in complex domain is (M(n)=?)? This problem relates to Hilbert’s 16th problem closely. Inequality M(3)≥7 have
文摘In this paper we study the variation of limit cycles around different foci when a coefficient in the equation of the quadratic differential system varies.
基金Supported by the National Natural Science Foundation of China(No.11371373 and 10831003)
文摘For the planar Z2-equivariant cubic systems having two elementary focuses, the characterization of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved. The necessary and sufficient conditions for the existence of the bi-center are obtained. On the basis of this work, in this paper, we show that under small Z2-equivariant cubic perturbations, this cubic system has at least 13 limit cycles with the scheme 1 6 ∪ 6.
