Math 2.0

Ordinary differential equations Basic notions Concept

Phase portrait

Builds on Phase planeTrajectoryEquilibrium point

Summary

A phase portrait of an autonomous system is a picture of a representative set of its trajectories in the phase space, usually the phase plane. Arrows on the curves show the direction of time. It shows the equilibrium points, the periodic orbits and the long-time behaviour of the solutions.

Remarks

The phase line is the phase portrait of a scalar autonomous equation. For the undamped harmonic oscillator x′′=−xx'' = -x, the phase portrait in the (x,x′)(x, x') plane consists of the origin and the circles around it.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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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…definitionLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionGrönwall's inequality, statement, no evidence levelGrönwall's inequalitystatementOrder of a differential equation, definition, no evidence levelOrder of a differenti…definitionSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionAutonomous system, definition, no evidence levelAutonomous systemdefinitionInitial value problem, definition, no evidence levelInitial value problemdefinitionLinear differential equation of order n, definition, no evidence levelLinear differential e…definitionEquilibrium point, definition, no evidence levelEquilibrium pointdefinitionSystem of linear differential equations, definition, no evidence levelSystem of linear diff…definitionUniqueness of solutions for Lipschitz vector fields, statement, no evidence levelUniqueness of solutio…statementMaximal solution, definition, no evidence levelMaximal solutiondefinitionFlow of a vector field, definition, no evidence levelFlow of a vector fielddefinitionTrajectory, definition, no evidence levelTrajectorydefinitionPhase plane, concept, no evidence levelPhase planeconceptPhase portrait, concept, no evidence levelPhase portraitconcept
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.
  • no evidence level
  1. Differential equation concept
  2. Function concept
  3. Interval concept
  4. Ordinary differential equation definition
  5. Lipschitz condition definition
  6. Grönwall's inequality statement
  7. Order of a differential equation definition
  8. Solution of an ordinary differential equation definition
  9. Autonomous system definition
  10. Initial value problem definition
  11. Linear differential equation of order n definition
  12. Equilibrium point definition
  13. System of linear differential equations definition
  14. Uniqueness of solutions for Lipschitz vector fields statement
  15. Maximal solution definition
  16. Flow of a vector field definition
  17. Trajectory definition
  18. Phase plane concept
  19. Phase portrait concept

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Phase_portrait, lead section
  2. Web https://en.wikipedia.org/wiki/Phase_portrait, Visualizing the behavior of ordinary differential equations
About this page
Tag0T3GK0XCBC
URIm2://0T3GK0XCBC
Kindconcept
WritingW1 stated
MSC34C05, 37C10
UsesPhase plane, Trajectory, Equilibrium point
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