Phase line

Summary

Let $f$ be a function on an interval of $\mathbb{R}$. The phase line of the scalar autonomous equation $x' = f(x)$ is a diagram of the line of states. It marks the equilibrium points, which are the zeros of $f$. It puts an arrow on each interval between them: up where $f > 0$ and down where $f < 0$. The arrows show the direction in which each solution moves.

Remarks

An equilibrium point with both arrows toward it is stable, or a sink. With both arrows away from it, it is unstable, or a source. With one arrow toward it and one away, it is semi-stable. For $f$ locally Lipschitz, a solution stays in one interval between equilibrium points and is monotone there. For the logistic equation $x' = x(1 - x)$, the point $0$ is a source and the point $1$ is a sink.

Explanations

L1 human

The phase line is a one-dimensional map of where the state can go. Arrows show whether the state increases or decreases, and the equilibrium points are the places where it stops.