Phase line

In mathematics, a phase line is a diagram which shows the behaviour of an autonomous ordinary differential equation. The term is also used in histogeographic maps and military maps to show some positional dependency or relation to the passage of time.

Mathematics
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.

Military usage
The term is also used in military terminology to refer to an imaginary line on a map used to show where specific phases of an operation will take place, hence the name. They are usually distinguished with different code names, varying by the particulars of the mission such as location and nature. By the same token, phase lines are frequently used after a battle or a military campaign by historians to describe what territory was controlled by which side at a given phase of the campaign.

Historical geography
Historical geographers use such techniques in a similar way, but written upon a longer time period. Dated phase lines on a map would indicate the growth or shrinkage of a great power of the era such as the expansion of the Roman Empire, the spread of Islam or Christianity. Other familiar phase line mapping most experience in late elementary or middle school classes would be the extent of various ice ages or the Ice Age consequential shrinkage or extension of continental landmasses down the continental shelves as the results of such climate changes.