Math 2.0

concept

Phase portrait

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.

Explanations

L1 human

A phase portrait is a map of all the possible motions of a system. One look shows where the system rests, where it cycles and where it goes in the long run.

What this node needs for W2
  • edge defines in: at least one definition (0 of 1)
  • edge instance_of in: at least two examples (0 of 2)
  • edge not_instance_of in: at least one non-example (0 of 1)
  • edge specializes both: the more general notions and the special cases (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 19 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

Sources

  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

What links here

No node links to this node yet.