How To Draw A Phase Line In Differential Equation?

A plot of $f(y)$ (left) and its phase line (correct). In this example, a and c are both sinks and b is a source.

In mathematics, a phase line is a diagram that shows the qualitative behaviour of an autonomous ordinary differential equation in a single variable, ${\tfrac {dy}{dx}}=f(y)$ . The stage line is the ane-dimensional form of the full general $north$ -dimensional phase space, and can be readily analyzed.

Diagram [edit]

A line, usually vertical, represents an interval of the domain of the derivative. The critical points (i.e., roots of the derivative ${\tfrac {dy}{dx}}$ , points $y$ such that $f(y)=0$ ) 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 management along the line (upward 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 stage line is identical in form to the line used in the first derivative test, other than being drawn vertically instead of horizontally, and the interpretation is virtually identical, with the same classification of critical points.

Examples [edit]

The simplest examples of a stage line are the petty stage lines, corresponding to functions $f(y)$ which exercise not change sign: if $f(y)=0$ , every point is a stable equilibrium ( $y$ does not modify); if $f(y)>0$ for all $y$ , and so $y$ is ever increasing, and if $f(y)<0$ then $y$ is e'er decreasing.

The simplest not-trivial examples are the exponential growth model/decay (1 unstable/stable equilibrium) and the logistic growth model (two equilibria, one stable, one unstable).

Nomenclature of critical points [edit]

A disquisitional signal can exist 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 indicate, it is stable (a sink): nearby solutions volition converge asymptotically 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 bespeak away from the critical point, it is unstable (a source): nearby solutions will diverge from the disquisitional point, and the solution is unstable under small perturbations, significant that if the solution is disturbed, information technology will not return to the solution.

Otherwise – if ane arrow points towards the disquisitional indicate, and one points away – it is semi-stable (a node): information technology 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 besides [edit]

Offset derivative examination, analog in elementary differential calculus
Phase plane, two-dimensional form
Stage infinite, $n$ -dimensional form

References [edit]

Equilibria and the Stage Line, past Mohamed Amine Khamsi, S.O.S. Math, final Update 1998-6-22
"The phase line and the graph of the vector field". math.bu.edu. Retrieved 2015-04-23 .