We study the properties of non-wandering points of the following scalar reaction-diffusion equation on the circle $S^1$,$u_t=u_{xx}+f(t,u,u_x),t>0,x\in S^1=\mathbb{R}/2\pi \mathbb{Z},$ where f is independent of t or T-periodic in t. Assume that the equation admits a compact global attractor. It is proved that, any non-wandering point is a limit point of the system (that is, it is a point in some ω-limit set). More precisely, in the autonomous case, it is proved that any non-wandering point is either a fixed point or generates a rotating wave on the circle. In the periodic case, it is proved that any non-wandering point is a periodic point or generates a rotating wave on a torus. In particular, if $f(t,u,-u_x)=f(t,u,u_x)$, then any non-wandering point is a fixed point in the autonomous case, and is a periodic point in the periodic case.

我们研究下列标量反应扩散方程在圆上的非游移点的性质 ${秒}^1$,${你}_{吨}={你}_{XX}+F(吨,你,{你}_X),吨>0,X\in {秒}^1=\mathbb{电阻}/2\pi \mathbb{Z},$其中˚F是独立的吨或Ť中为周期吨。假设方程允许一个紧凑的全局吸引子。证明，任何非游移点都是系统的一个极限点（即它是某个ω-极限集中的一个点）。更准确地说，在自治的情况下，证明了任何非漂移点要么是固定点，要么在圆上产生旋转波。在周期情况下，证明任何非游移点都是周期点或在环面上产生旋转波。特别地，如果$F(吨,你,-{你}_X)=F(吨,你,{你}_X)$，则任意非游移点在自治情况下为不动点，在周期情况下为周期点。