Extended quasi-homogeneous polynomial system in $${\mathbb{R}^{3}}$$

Abstract

Abstract. In this paper we define an extended quasi-homogeneous polynomial system dx/dt = Q = Q1 + Q2 + · · · + Q δ , where Qi are some 3-dimensional quasi-homogeneous vectors with weight α and degree i, i = 1, . . . , δ. Firstly we investigate the limit set of trajectory of this system. Secondly let QT be the projective vector field of Q. We show that if δ ≤ 3 and the number of closed orbits of QT is known, then an upper bound for the number of isolated closed orbits of the system is obtained. Moreover this upper bound is sharp for δ = 3. As an application, we show that a 3-dimensional polynomial system of degree 3 (resp. 5) admits 26 (resp. 112) isolated closed orbits. Finally, we prove that a 3-dimensional Lotka-Volterra system has no isolated closed orbits in the first octant if it is extended quasi-homogeneous.

Bifurcation of a class of extended quasi-homogeneous polynomial differential systems

“…. , δ − 1) 的 (α 1 , α 2 )-拟齐次多项式, 则称 (1.2) 为拟齐次多项式向量 场 Q δ 的扩展拟齐次系统, 参见文献 [1]. 至今, 已有许多文献研究了齐次和拟齐次多项式微分系统.…”

“…文献 [6] 证明了 4 次拟齐次且非齐次系统有 2 类中心, 并给出了它们的拓扑相图. 文 献 [1,7] 研究了 3 维扩展拟齐次系统的孤立闭轨个数的上界. 文献 [8] 研究了一类特殊的平面多项式 系统的极限环, 其向量场由 2 个具有特定权重和权指的拟齐次多项式构成, 可以将它转化为 Abel 方 程研究.…”