Math 2.0

Ordinary differential equations Concept

Classification of equilibria of planar linear systems

Builds on Homogeneous linear system with constant coefficientsPhase planeStability of linear systems via eigenvalues

Summary

Let AA be a real 2×22 \times 2 matrix with det⁡A≠0\det A \neq 0, let τ=tr⁡A\tau = \operatorname{tr} A and Δ=det⁡A\Delta = \det A, and consider x′=Axx' = A x in the phase plane. The origin is a saddle if Δ<0\Delta < 0. It is a node if Δ>0\Delta > 0 and τ2−4Δ≥0\tau^2 - 4 \Delta \ge 0, and a spiral, or focus, if τ≠0\tau \neq 0 and τ2−4Δ<0\tau^2 - 4 \Delta < 0. It is a centre if τ=0\tau = 0 and Δ>0\Delta > 0. A node or a spiral is stable for τ<0\tau < 0 and unstable for τ>0\tau > 0.

Remarks

The eigenvalues are the roots of λ2−τλ+Δ=0\lambda^2 - \tau \lambda + \Delta = 0. A saddle has real eigenvalues of opposite signs. A node has real eigenvalues of the same sign. A spiral or a centre has complex conjugate eigenvalues. The case τ2=4Δ\tau^2 = 4 \Delta gives a repeated eigenvalue. The origin is then a star node or a degenerate node. The trace and the determinant do not change under a change of basis A↦C−1ACA \mapsto C^{-1} A C. So the classification depends only on the system, not on the coordinates. The damped harmonic oscillator gives a stable spiral when it is underdamped and a stable node when it is overdamped.

Reading path

To understand this page from the ground up, read these 24 pages in order. The foundations come first.

Show the path as a graph
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…statementHomogeneous linear system with constant coefficients, definition, no evidence levelHomogeneous linear sy…definitionMaximal solution, definition, no evidence levelMaximal solutiondefinitionMatrix exponential as the solution of a linear system, statement, no evidence levelMatrix exponential as…statementFlow of a vector field, definition, no evidence levelFlow of a vector fielddefinitionLyapunov stability, definition, no evidence levelLyapunov stabilitydefinitionTrajectory, definition, no evidence levelTrajectorydefinitionAsymptotic stability, definition, no evidence levelAsymptotic stabilitydefinitionPhase plane, concept, no evidence levelPhase planeconceptStability of linear systems via eigenvalues, statement, no evidence levelStability of linear s…statementClassification of equilibria of planar linear systems, concept, no evidence levelClassification of equ…concept
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. Homogeneous linear system with constant coefficients definition
  16. Maximal solution definition
  17. Matrix exponential as the solution of a linear system statement
  18. Flow of a vector field definition
  19. Lyapunov stability definition
  20. Trajectory definition
  21. Asymptotic stability definition
  22. Phase plane concept
  23. Stability of linear systems via eigenvalues statement
  24. Classification of equilibria of planar linear systems concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Stability_theory, Stability of fixed points in 2D
  2. Web https://en.wikipedia.org/wiki/Stability_theory, Classification of stability types
About this page
Tag0N7ATVF2X8
URIm2://0N7ATVF2X8
Kindconcept
WritingW0 stub
MSC34C05, 34D20
UsesHomogeneous linear system with constant coefficients, Phase plane, Stability of linear systems via eigenvalues

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

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

What links here

No page links to this page yet.