Math 2.0

concept

Phase plane

Summary

The phase plane of a planar autonomous system x′=f(x,y)x' = f(x, y), y′=g(x,y)y' = g(x, y) is the plane with coordinates (x,y)(x, y), the two state variables. Each solution traces a curve in this plane, its trajectory. The vector (f(x,y),g(x,y))(f(x, y), g(x, y)) is tangent to the trajectory through (x,y)(x, y). It is the two-dimensional case of the phase space.

Remarks

A second-order equation x′′=h(x,x′)x'' = h(x, x') gives a planar system with y=x′y = x'. Its phase plane has the position and the velocity as coordinates. For a linear system (x,y)′=A(x,y)(x, y)' = A (x, y) with a 2×22 \times 2 real matrix AA, the eigenvalues of AA determine the shape of the trajectories. See the classification of planar linear systems.

Explanations

L1 human

The phase plane shows the two state variables at once, without the time axis. Each point is a state, and the system moves from state to state along a curve.

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  • 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)

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To read this node, read these 17 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…definitionSystem 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 planeconcept
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.
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  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. System of linear differential equations definition
  13. Uniqueness of solutions for Lipschitz vector fields statement
  14. Maximal solution definition
  15. Flow of a vector field definition
  16. Trajectory definition
  17. Phase plane concept

What depends on this

4 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Phase_plane, lead section
  2. Web https://en.wikipedia.org/wiki/Phase_plane, Example of a linear system

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