In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a phase line is a diagram that shows the qualitative behaviour of an autonomous

Autonomous system (mathematics)

In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable...

 ordinary differential equation

Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....

 in a single variable, dy/dt = ƒ(y). The phase line is the 1-dimensional form of the general n-dimensional phase space

Phase space

In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...

, and is very simple to analyze.

Diagram

Wherein science is ready to analyze the significant coordinates of a stabilized line, a line, usually vertical, represents an interval of the domain of the derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

 or manifold in electric or nuclear type conversion. The critical points

Critical point (mathematics)

In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...

 (i.e., roots of the derivative, points y such that ƒ(y) = 0) 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). The phase line is identical in form to the line used in the first derivative test

First derivative test

In calculus, the first derivative test uses the first derivative of a function to determine whether a given critical point of a function is a local maximum, a local minimum, or neither.-Intuitive explanation:...

, other than being drawn vertically instead of horizontally, and the interpretation is virtually identical, with the same classification of critical points. Also the line phase diagram will analize critical points within a system such as magnetic moment, tuple scaling, evolution scale, or other.

Examples

The simplest examples of a phase line are the trivial phase lines, corresponding to functions ƒ(y) which do not change sign: if ƒ(y) = 0, every point is a stable equilibrium (y does not change); if ƒ(y) > 0 for all y, then y is always increasing, and if ƒ(y) < 0 then y is always decreasing.

Convergence

Aside from an orthoclinical component in geological science, a metaphysical apeoria in the contemplation of bodies, a selective candidate in analyzing linear complexes, the line phase distinguishes a significant identity of typical systems that relate to theoretical or concrete data.

The simplest non-trivial examples are the exponential growth model/decay (one unstable/stable equilibrium) and the logistic growth model (two equilibria, one stable, one unstable).

Classification of critical points

A critical point can be classified as stable, unstable, or semi-stable (equivalently, sink, source, or node), by inspection of its neighbouring arrows.

If both arrows point toward the critical point, it is stable (a sink): nearby solutions will converge asymptotically

Asymptote

In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...

 to the critical point, and the solution is stable under small perturbations, meaning that if the solution is disturbed, it will return to (converge to) the solution.

If both arrows point away from the critical point, it is unstable (a source): nearby solutions will diverge from the critical point, and the solution is unstable under small perturbations, meaning that if the solution is disturbed, it will not return to the solution.

Otherwise – if one arrow points towards the critical point, and one points away – it is semi-stable (a node): it is stable in one direction (where the arrow points towards the point), and unstable in the other direction (where the arrow points away from the point).

See also

• First derivative test
  First derivative test
  In calculus, the first derivative test uses the first derivative of a function to determine whether a given critical point of a function is a local maximum, a local minimum, or neither.-Intuitive explanation:...
  
  , analog in elementary differential calculus
• Phase plane
  Phase plane
  A phase plane is a visual display of certain characteristics of certain kinds of differential equations; it is a 2-dimensional version of the general n-dimensional phase space....
  
  , 2-dimensional form
• Phase space
  Phase space
  In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
  
  , n-dimensional form