In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a phase line is a diagram that shows the qualitative behaviour of an
autonomous
In developmental psychology and moral, political, and bioethical philosophy, autonomy, from , ''autonomos'', from αὐτο- ''auto-'' "self" and νόμος ''nomos'', "law", hence when combined understood to mean "one who gives oneself one's ow ...
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
in a single variable,
. The phase line is the 1-dimensional form of the general
-dimensional
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
, and can be readily analyzed.
Diagram
A line, usually vertical, represents an interval of the domain of the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
. The
critical points (i.e.,
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of the derivative
, points
such that
) 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
In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about ...
, other than being drawn vertically instead of horizontally, and the interpretation is virtually identical, with the same classification of critical points.
Examples
The simplest examples of a phase line are the trivial phase lines, corresponding to functions
which do not change sign: if
, every point is a stable equilibrium (
does not change); if
for all
, then
is always increasing, and if
then
is always decreasing.
The simplest non-trivial examples are the
exponential growth model
Population growth is the increase in the number of people in a population or dispersed group. Actual global human population growth amounts to around 83 million annually, or 1.1% per year. The global population has grown from 1 billion in 1800 to ...
/decay (one unstable/stable equilibrium) and the
logistic growth model
A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with equation
f(x) = \frac,
where
For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is obtained, with the ...
(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
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
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
In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about ...
, analog in elementary differential calculus
*
Phase plane
In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being the values of the two sta ...
, 2-dimensional form
*
Phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
,
-dimensional form
References
Equilibria and the Phase Line by Mohamed Amine Khamsi, S.O.S. Math, last Update 1998-6-22
*
{{refend
Ordinary differential equations