Math 2.0

Ordinary differential equations Stability Concept

Linearisation at an equilibrium

Builds on Equilibrium pointHomogeneous linear system with constant coefficientsStability of linear systems via eigenvalues

Summary

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, let f:U→ℝnf : U \to \mathbb{R}^n be continuously differentiable, and let x∗x^* be an equilibrium point of x′=f(x)x' = f(x). Let A=Df(x∗)A = Df(x^*) be the Jacobian matrix of ff at x∗x^*. The linearisation of the system at x∗x^* is the linear system y′=Ayy' = A y, where yy stands for x−x∗x - x^*. It comes from the first-order Taylor expansion f(x)=A(x−x∗)+o(∥x−x∗∥)f(x) = A (x - x^*) + o(\|x - x^*\|).

Remarks

If each eigenvalue of AA has negative real part, then x∗x^* is asymptotically stable for the nonlinear system. If one eigenvalue has positive real part, then x∗x^* is unstable. If some eigenvalue has zero real part and none has positive real part, the linearisation does not decide stability. For x′=−x3x' = -x^3 and x′=x3x' = x^3, the linearisation at 00 is y′=0y' = 0 in both cases, but the first is stable and the second is not. Near a hyperbolic equilibrium, the Hartman–Grobman theorem says that the linearisation describes the phase portrait up to a homeomorphism.

Reading path

To understand this page from the ground up, read these 22 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…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 stabilitydefinitionAsymptotic stability, definition, no evidence levelAsymptotic stabilitydefinitionStability of linear systems via eigenvalues, statement, no evidence levelStability of linear s…statementLinearisation at an equilibrium, concept, no evidence levelLinearisation at an e…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. Asymptotic stability definition
  21. Stability of linear systems via eigenvalues statement
  22. Linearisation at an equilibrium concept

Where this leads

10 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Linearization, Stability analysis
  2. Web https://en.wikipedia.org/wiki/Stability_theory, Non-linear autonomous systems
About this page
Tag09ZWYRQ41J
URIm2://09ZWYRQ41J
Aliaseslinearization
Kindconcept
WritingW0 stub
MSC34D20, 34C05
UsesEquilibrium point, Homogeneous linear system with constant coefficients, Stability of linear systems via eigenvalues

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