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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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. 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).