Phase line

The phase line of a scalar autonomous differential equation $\dot{x}=f(x)$ records its equilibrium points and the direction of motion between them. Zeros of $f$ mark equilibria; $f>0$ gives rightward arrows and $f<0$ gives leftward arrows. Local uniqueness prevents a nonconstant solution from crossing an equilibrium.

For example, $\dot{x}=x(1-x)$ has equilibria zero and one. Arrows point left for $x<0$, right for $0<x<1$ and left for $x>1$. Thus zero repels nearby states and one attracts them. Every solution starting in $(0,1)$ stays there and increases toward one: its limit cannot be an interior point where $f$ is positive.

This argument gives qualitative dynamics without solving the equation. Inward arrows on both sides indicate attraction, outward arrows indicate repulsion, and arrows in the same direction describe one-sided stability. The latter occurs at zero for $\dot{x}=x^2$: negative initial states approach it, while positive initial states move away.