Math 2.0

Ordinary differential equations Concept

Phase line

Builds on Autonomous systemEquilibrium point

Summary

Let ff be a function on an interval of ℝ\mathbb{R}. The phase line of the scalar autonomous equation x′=f(x)x' = f(x) is a diagram of the line of states. It marks the equilibrium points, which are the zeros of ff. It puts an arrow on each interval between them: up where f>0f > 0 and down where f<0f < 0. The arrows show the direction in which each solution moves.

Remarks

An equilibrium point with both arrows toward it is stable, or a sink. With both arrows away from it, it is unstable, or a source. With one arrow toward it and one away, it is semi-stable. For ff locally Lipschitz, a solution stays in one interval between equilibrium points and is monotone there. For the logistic equation x′=x(1−x)x' = x(1 - x), the point 00 is a source and the point 11 is a sink.

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Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionAutonomous system, definition, no evidence levelAutonomous systemdefinitionEquilibrium point, definition, no evidence levelEquilibrium pointdefinitionPhase line, concept, no evidence levelPhase lineconcept
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Differential equation concept
  2. Function concept
  3. Interval concept
  4. Ordinary differential equation definition
  5. Solution of an ordinary differential equation definition
  6. Autonomous system definition
  7. Equilibrium point definition
  8. Phase line concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Phase_line_(mathematics), Diagram
  2. Web https://en.wikipedia.org/wiki/Phase_line_(mathematics), Classification of critical points
About this page
Tag081QSEAGEY
URIm2://081QSEAGEY
Kindconcept
WritingW0 stub
MSC34C05, 34A34
SpecializesPhase portrait
UsesAutonomous system, Equilibrium point
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