This paper considers the problems of determining center or focus and isochronous centers for the planar quasi-analytic systems. Two recursive formulas to determine the focal values and period constants are given. The ...This paper considers the problems of determining center or focus and isochronous centers for the planar quasi-analytic systems. Two recursive formulas to determine the focal values and period constants are given. The convergence of first integral near the center is proved. Using the general results to quasi-quadratic systems, the problem of the isochronous center of the origin is completely solved.展开更多
In this paper, we study the limit cycles bifurcations of four fine focuses in Z4-equivariant vector fields and the problems that its four singular points can be centers and isochronous centers at the same time. By com...In this paper, we study the limit cycles bifurcations of four fine focuses in Z4-equivariant vector fields and the problems that its four singular points can be centers and isochronous centers at the same time. By computing the Liapunov constants and periodic constants carefully, we show that for a certain Z4-equivariant quintic systems, there are four fine focuses of five order and five limit cycles can bifurcate from each, we also find conditions of center and isochronous center for this system. The process of proof is algebraic and symbolic by using common computer algebra soft such as Mathematica, the expressions after being simplified in this paper are simple relatively. Moreover, what is worth mentioning is that the result of 20 small limit cycles bifurcating from several fine focuses is good for Z4-equivariant quintic system and the results where multiple singular points become isochronous centers at the same time are less in published references.展开更多
We study isochronous centers of two classes of planar systems of ordinary differential equations. For the first class which is the Lienard systems of the form x=y-F(x), y=-g(x) with a center at the origin, we prove th...We study isochronous centers of two classes of planar systems of ordinary differential equations. For the first class which is the Lienard systems of the form x=y-F(x), y=-g(x) with a center at the origin, we prove that if g is isochronous (see Definition 1.1), then the center is isochronous if and only if F≡0. For the second class which is the Hamiltonian systems of the form i=-g(y), y=f(x) with a center at the origin, we prove that if / or g is isochronous, then the center is isochronous if and only if the other is also isochronous.展开更多
基金the National Natural Science Foundation of China (10671179 and 10771196)the Natural Science Foundation of Yunnan Province (2005A0092M)
文摘This paper considers the problems of determining center or focus and isochronous centers for the planar quasi-analytic systems. Two recursive formulas to determine the focal values and period constants are given. The convergence of first integral near the center is proved. Using the general results to quasi-quadratic systems, the problem of the isochronous center of the origin is completely solved.
基金Partially supported by National Natural Science Foundation of China (Grant No. 10771196)the Research Fund of Hunan Provincial Education Department (Grant No. 09A082)Hunan Provincial Natural Science Foundation (Grant No. 10JJ5046)
文摘In this paper, we study the limit cycles bifurcations of four fine focuses in Z4-equivariant vector fields and the problems that its four singular points can be centers and isochronous centers at the same time. By computing the Liapunov constants and periodic constants carefully, we show that for a certain Z4-equivariant quintic systems, there are four fine focuses of five order and five limit cycles can bifurcate from each, we also find conditions of center and isochronous center for this system. The process of proof is algebraic and symbolic by using common computer algebra soft such as Mathematica, the expressions after being simplified in this paper are simple relatively. Moreover, what is worth mentioning is that the result of 20 small limit cycles bifurcating from several fine focuses is good for Z4-equivariant quintic system and the results where multiple singular points become isochronous centers at the same time are less in published references.
文摘We study isochronous centers of two classes of planar systems of ordinary differential equations. For the first class which is the Lienard systems of the form x=y-F(x), y=-g(x) with a center at the origin, we prove that if g is isochronous (see Definition 1.1), then the center is isochronous if and only if F≡0. For the second class which is the Hamiltonian systems of the form i=-g(y), y=f(x) with a center at the origin, we prove that if / or g is isochronous, then the center is isochronous if and only if the other is also isochronous.
