275:) 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
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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).
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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
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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.
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A critical point can be classified as stable, unstable, or semi-stable (equivalently, sink, source, or node), by inspection of its neighbouring arrows.
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If both arrows point toward the critical point, it is stable (a sink): nearby solutions will converge
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The simplest examples of a phase line are the trivial phase lines, corresponding to functions
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A line, usually vertical, represents an interval of the domain of the
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Diagram used to analyze autonomous ordinary differential equations
597:, by Mohamed Amine Khamsi, S.O.S. Math, last Update 1998-6-22
137:. The phase line is the 1-dimensional form of the general
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is a diagram that shows the qualitative behaviour of an
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602:"The phase line and the graph of the vector field"
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508:/decay (one unstable/stable equilibrium) and the
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512:(two equilibria, one stable, one unstable).
504:The simplest non-trivial examples are the
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351:, every point is a stable equilibrium (
130:{\displaystyle {\tfrac {dy}{dx}}=f(y)}
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516:Classification of critical points
213:{\displaystyle {\tfrac {dy}{dx}}}
627:Ordinary differential equations
161:, and can be readily analyzed.
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81:ordinary differential equation
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595:Equilibria and the Phase Line
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474:{\displaystyle f(y)<0}
399:{\displaystyle f(y)>0}
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344:{\displaystyle f(y)=0}
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583:-dimensional form
576:{\displaystyle n}
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175:critical points
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604:. math.bu.edu
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157:-dimensional
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606:. Retrieved
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64:is a source.
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561:Phase space
555:Phase plane
159:phase space
70:mathematics
608:2015-04-23
588:References
240:such that
171:derivative
78:autonomous
74:phase line
23:A plot of
220:, points
621:Category
543:See also
406:for all
283:Examples
426:, then
177:(i.e.,
165:Diagram
173:. The
481:then
179:roots
466:<
391:>
72:, a
56:and
533:not
68:In
623::
563:,
611:.
571:n
489:y
469:0
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336:=
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298:(
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263:0
260:=
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251:(
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196:y
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145:n
125:)
122:y
119:(
116:f
113:=
106:x
103:d
98:y
95:d
62:b
58:c
54:a
40:)
37:y
34:(
31:f
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