In this paper,we use the canonical forms of homogeneous polynomials of degree 3 to study the global properties of cubic systems =x+P<sub>3</sub>（x,y）,=y+Q<sub>3</sub>（x,y）（0.1） where P<...In this paper,we use the canonical forms of homogeneous polynomials of degree 3 to study the global properties of cubic systems =x+P<sub>3</sub>（x,y）,=y+Q<sub>3</sub>（x,y）（0.1） where P<sub>3</sub> and Q<sub>3</sub> are homogeneous polynomials of degree 3 in x,y.Through this work,we draw an overall outline of such systems.展开更多
The existence and uniqueness of limit cycle for the E 1 3 type of cubic systems with two integral straight lines has been studied in this paper. It is found that the system has no limit cycle when the two int...The existence and uniqueness of limit cycle for the E 1 3 type of cubic systems with two integral straight lines has been studied in this paper. It is found that the system has no limit cycle when the two integral straight lines intersect each other; it has a unique limit cycle when the two integral straight lines are paralleled. The sufficient and necessary conditions are also given to guarantee the existence of the unique limit cycle.展开更多
In this paper, we consider a system of two cubic quasi-linear Klein-Gordon equations with different masses for small, smooth, compactly supported Cauchy data in one space dimension. We show that such a system has glob...In this paper, we consider a system of two cubic quasi-linear Klein-Gordon equations with different masses for small, smooth, compactly supported Cauchy data in one space dimension. We show that such a system has global existence when the nonlinearities satisfy a convenient null condition. Our results extend the global existence proved by Sunagawa recently under the non-resonance assumption to that under the resonance assumption.展开更多
For cubic differential systems with two homogeneous invariant straight lines and one invariant conic, it is proved that a singular point with pure imaginary eigenvalues (a weak focus) is a centre if and only if the fi...For cubic differential systems with two homogeneous invariant straight lines and one invariant conic, it is proved that a singular point with pure imaginary eigenvalues (a weak focus) is a centre if and only if the first two Lyapunov quantities Lj , j = 1, 2 vanish.展开更多
In this paper, by using model-theoretic methods, it is shown that some systems of unsolved cubic diophantine equations in number theory can have solutions in certain inductive extension rings of the ring I of rational...In this paper, by using model-theoretic methods, it is shown that some systems of unsolved cubic diophantine equations in number theory can have solutions in certain inductive extension rings of the ring I of rational integers. These inductive rings are not fields, and every element of them is a sum of 4 cubes and a sum of 3 squares. Also some of them satisfy the Goldbach conjecture and some others don＇t.展开更多
基金Supported by the National Natural Science Foundation of China,No.19371069
文摘In this paper,we use the canonical forms of homogeneous polynomials of degree 3 to study the global properties of cubic systems =x+P<sub>3</sub>（x,y）,=y+Q<sub>3</sub>（x,y）（0.1） where P<sub>3</sub> and Q<sub>3</sub> are homogeneous polynomials of degree 3 in x,y.Through this work,we draw an overall outline of such systems.
文摘The existence and uniqueness of limit cycle for the E 1 3 type of cubic systems with two integral straight lines has been studied in this paper. It is found that the system has no limit cycle when the two integral straight lines intersect each other; it has a unique limit cycle when the two integral straight lines are paralleled. The sufficient and necessary conditions are also given to guarantee the existence of the unique limit cycle.
文摘In this paper, we consider a system of two cubic quasi-linear Klein-Gordon equations with different masses for small, smooth, compactly supported Cauchy data in one space dimension. We show that such a system has global existence when the nonlinearities satisfy a convenient null condition. Our results extend the global existence proved by Sunagawa recently under the non-resonance assumption to that under the resonance assumption.
文摘For cubic differential systems with two homogeneous invariant straight lines and one invariant conic, it is proved that a singular point with pure imaginary eigenvalues (a weak focus) is a centre if and only if the first two Lyapunov quantities Lj , j = 1, 2 vanish.
基金Supported by NNSF(No. 19931020, No. 10001006 and No. 60273015)of China
文摘In this paper, by using model-theoretic methods, it is shown that some systems of unsolved cubic diophantine equations in number theory can have solutions in certain inductive extension rings of the ring I of rational integers. These inductive rings are not fields, and every element of them is a sum of 4 cubes and a sum of 3 squares. Also some of them satisfy the Goldbach conjecture and some others don＇t.
