Phase line

From Wikipedia, the free encyclopedia

In mathematics, a phase line is a diagram which shows the behaviour of an autonomous ordinary differential equation.

A line, usually vertical, represents an interval of the domain of the derivative. The critical points (i.e., roots of the derivative) are indicated, and the intervals between the critical points have their signs indicated with arrows: an interval over which the derivative is positive has an arrow pointing in the positive direction along the line (up or right), and an interval over which the derivative is negative has an arrow pointing in the negative direction along the line (down or left).

A critical point can be classified as stable, semi-stable, or unstable, by inspection of its neighbouring arrows. If both arrows point toward the critical point, it is stable, and nearby solutions will converge asymptotically to the critical point. If both arrows point away from the critical point, it is unstable. Otherwise, it is semi-stable.