Math 2.0

Ordinary differential equations Definition

Structural stability

Builds on Vector fieldHomeomorphismTrajectoryContinuously differentiable function

Definition

Let MM be a compact smooth manifold without boundary, such as a circle, a sphere or a torus. Let 𝔛1(M)\mathfrak{X}^1(M) be the space of continuously differentiable vector fields on MM, with the C1C^1 topology. Two vector fields XX and YY on MM are topologically equivalent if there is a homeomorphism h ⁣:M→Mh \colon M \to M with two properties. It maps each trajectory of XX onto a trajectory of YY. It preserves the direction of time along trajectories. A vector field X∈𝔛1(M)X \in \mathfrak{X}^1(M) is structurally stable if it has a neighbourhood 𝒩\mathcal{N} in 𝔛1(M)\mathfrak{X}^1(M) such that every Y∈𝒩Y \in \mathcal{N} is topologically equivalent to XX.

Remarks

Example: on the circle M=ℝ/2πℤM = \R / 2\pi\Z, the field X(θ)=sin⁡θX(\theta) = \sin \theta has the two zeros 00 and π\pi, with X′(0)=1X'(0) = 1 and X′(π)=−1X'(\pi) = -1. A field YY that is C1C^1-close to XX still has exactly two zeros, near 00 and π\pi, with derivatives of the same signs. So YY has one repelling and one attracting rest point joined by two arcs, and a homeomorphism of the circle carries the trajectories of XX to those of YY. Hence XX is structurally stable.

Non-example: X(θ)=1−cos⁡θX(\theta) = 1 - \cos \theta has the single zero 00. For ε>0\varepsilon > 0, the field 1−cos⁡θ+ε1 - \cos \theta + \varepsilon has no zero, and it is C1C^1-close to XX for small ε\varepsilon. A homeomorphism maps rest points to rest points, so these fields are not equivalent to XX, and XX is not structurally stable.

For flows on a compact surface, structural stability is closely tied to hyperbolicity of the rest points and periodic orbits and to the absence of trajectories joining saddles. For a region in the plane, the definition is adapted by asking the vector field to point inward on the boundary.

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Continuously differentiable function, concept, no evidence levelContinuously differen…conceptDifferential equation, concept, no evidence levelDifferential equationconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHomeomorphism, concept, no evidence levelHomeomorphismconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionVector-valued function, concept, no evidence levelVector-valued functionconceptLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionGrönwall's inequality, statement, no evidence levelGrönwall's inequalitystatementVector field, concept, no evidence levelVector fieldconceptSolution 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 problemdefinitionUniqueness 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 levelTrajectorydefinitionStructural stability, definition, no evidence levelStructural stabilitydefinition
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. Continuously differentiable function concept
  2. Differential equation concept
  3. Euclidean space object
  4. Function concept
  5. Homeomorphism concept
  6. Interval concept
  7. Ordinary differential equation definition
  8. Vector-valued function concept
  9. Lipschitz condition definition
  10. Grönwall's inequality statement
  11. Vector field concept
  12. Solution of an ordinary differential equation definition
  13. Autonomous system definition
  14. Initial value problem definition
  15. Uniqueness of solutions for Lipschitz vector fields statement
  16. Maximal solution definition
  17. Flow of a vector field definition
  18. Trajectory definition
  19. Structural stability definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Structural_stability
About this page
TagKQX43WB57R
URIm2://KQX43WB57R
Kinddefinition
WritingW0 stub
MSC34D30, 37C20
DefinesStructural stability
UsesVector field, Homeomorphism, Trajectory, Continuously differentiable function
Changed2 November 2026

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