Math 2.0

Dynamical systems and ergodic theory Smooth and hyperbolic dynamics Theorem

Hartman–Grobman theorem

Builds on Hyperbolic equilibrium pointLinearisation at an equilibriumFlow of a vector fieldMatrix exponential as the solution of a linear system

Statement

Let UU be an open subset of ℝn\mathbb{R}^n, let f:U→ℝnf : U \to \mathbb{R}^n be continuously differentiable. Let φ\varphi be the flow of x′=f(x)x' = f(x), and let x∗∈Ux^* \in U be a hyperbolic equilibrium point. Let A=Df(x∗)A = Df(x^*). Then there are open neighbourhoods V⊆UV \subseteq U of x∗x^* and W⊆ℝnW \subseteq \mathbb{R}^n of 00, and a homeomorphism h:V→Wh : V \to W with h(x∗)=0h(x^*) = 0, as follows. For each x∈Vx \in V and each t∈ℝt \in \mathbb{R} such that φs(x)∈V\varphi_s(x) \in V for all ss between 00 and tt, h(φt(x))=etAh(x)h(\varphi_t(x)) = e^{tA} h(x).

Remarks

So near x∗x^* the flow is topologically conjugate to the flow of the linearisation y′=Ayy' = A y, given by the matrix exponential. The map hh sends trajectories to trajectories and keeps the direction of time. The map hh is in general only continuous, not differentiable. The hypothesis of hyperbolicity is necessary. For x′=−x3x' = -x^3 and x′=x3x' = x^3 on ℝ\mathbb{R}, the linearisation at 00 is the same, but the two phase portraits are not conjugate.

History

The theorem is named after Philip Hartman and David M. Grobman. It is also called the linearization theorem.

Reading path

To understand this page from the ground up, read these 24 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…conceptHyperbolic equilibrium point, definition, no evidence levelHyperbolic equilibriu…definitionHartman–Grobman theorem, statement, no evidence levelHartman–Grobman theor…statement
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
  23. Hyperbolic equilibrium point definition
  24. Hartman–Grobman theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Hartman%E2%80%93Grobman_theorem, Main theorem
About this page
Tag0S8WX8A4V9
URIm2://0S8WX8A4V9
Aliaseslinearization-theorem
Kindstatement, theorem
WritingW1 stated
MSC37C15, 34C20
StatusThis copy of the site has no build data. Run m2 status 0S8WX8A4V9.
UsesHyperbolic equilibrium point, Linearisation at an equilibrium, Flow of a vector field, Matrix exponential as the solution of a linear system
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