The well-known Poincaré-Bendixson theorem tells us that the structure of the omega-limit sets of planar autonomous differential equations can be described by fixed points, limit cycles, or finite number of fixed points together with homoclinic and heteroclinic orbits connecting them. However, this is very different for planar nonautonomous differential equations. In this paper, we study the omega-limit sets of three classes of planar nonautonomous differential equations, and their corresponding dynamical behavior. (i) The first type is linear, the omega-limit set is an annulus, the Lyapunov exponents are zero, there is no periodic orbit except for one fixed point, and the orbits are transitive in the omega-limit sets, implying the existence of infinitely many transitive components depending on initial conditions. (ii) The second type has butterfly-like omega-limit sets, two zero Lyapunov exponents, and neither periodic orbits nor sensitive dependence on initial conditions. (iii) The last type has a unique omega-limit set (a global attractor) independent on initial conditions, and has two negative Lyapunov exponents, where the omega-limit set is an annulus or a subset (homeomorphic to a disk) of an annulus for different parameter regions.

著名的庞加莱-本迪克森（Poincaré-Bendixson）定理告诉我们，平面自治微分方程的欧米伽极限集的结构可以用固定点，极限环或有限个固定点以及连接它们的等斜轨道和异斜轨道描述。但是，这对于平面非自治微分方程是非常不同的。在本文中，我们研究了三类平面非自治微分方程的ω-极限集及其对应的动力学行为。（i）第一种是线性的，欧米伽极限集是一个环，李雅普诺夫指数是零，除了一个固定点外没有周期轨道，并且这些轨道在欧米伽极限集内是可传递的，这意味着存在取决于初始条件的无限多个传递成分。（ii）第二种类型具有类似蝴蝶的omega-limit集，两个零Lyapunov指数，并且既没有周期性的轨道，也没有对初始条件的敏感依赖。（iii）最后一种类型具有独立于初始条件的唯一欧米伽极限集（全局吸引子），并具有两个负Lyapunov指数，其中欧米伽极限集是环的圆环或子集（同胚圆盘）。不同参数区域的环空。