Math 2.0

Ordinary differential equations Definition

Hyperbolic equilibrium point

Builds on Equilibrium pointLinearisation at an equilibrium

Definition

Let UU be an open subset of ℝn\mathbb{R}^n, let f:U→ℝnf : U \to \mathbb{R}^n be continuously differentiable, and let x∗∈Ux^* \in U be an equilibrium point of x′=f(x)x' = f(x). The point x∗x^* is hyperbolic if no eigenvalue of the Jacobian matrix Df(x∗)Df(x^*) has real part 00.

Remarks

A saddle, a node and a spiral of a planar linear system are hyperbolic. A centre is not. For the pendulum x′=yx' = y, y′=−sin⁡xy' = -\sin x, the equilibrium (π,0)(\pi, 0) is hyperbolic, with eigenvalues ±1\pm 1. The equilibrium (0,0)(0, 0) is not, with eigenvalues ±i\pm i. Near a hyperbolic equilibrium, the linearisation describes the system qualitatively, by the Hartman–Grobman theorem. If all eigenvalues have negative real part, the point is a sink. If all have positive real part, it is a source. Otherwise it is a saddle.

Reading path

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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…definition
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.
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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. 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

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Hyperbolic_equilibrium_point, Flows
About this page
Tag0C5PREPZWG
URIm2://0C5PREPZWG
Aliaseshyperbolic-fixed-point
Kinddefinition
WritingW0 stub
MSC34C05, 37C75
DefinesHyperbolic equilibrium point
UsesEquilibrium point, Linearisation at an equilibrium

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