Math 2.0

Ordinary differential equations Definition

Center manifold

Builds on Autonomous systemEquilibrium pointJacobian matrixInvariant set

Definition

Let UU be an open subset of ℝn\R^n and let f ⁣:U→ℝnf \colon U \to \R^n be of class CkC^k with k≥1k \ge 1. Let φ\varphi be the flow of the autonomous system x′=f(x)x' = f(x). Let x0∈Ux_0 \in U be an equilibrium point. Let EcE^c be the sum of the generalised eigenspaces of the Jacobian matrix Df(x0)Df(x_0) for eigenvalues with real part 00. A local center manifold of x0x_0 is a set WcW^c with these properties:

  1. WcW^c is an embedded C1C^1 submanifold of an open neighbourhood V⊆UV \subseteq U of x0x_0, it contains x0x_0, and its tangent space at x0x_0 is EcE^c.
  2. WcW^c is locally invariant: if x∈Wcx \in W^c and φs(x)∈V\varphi_s(x) \in V for all ss between 00 and tt, then φt(x)∈Wc\varphi_t(x) \in W^c.

Remarks

The center manifold theorem states that a local center manifold exists and can be chosen of class CkC^k, but it is not unique in general. Example: for x′=x2x' = x^2, y′=−yy' = -y at the origin, EcE^c is the xx-axis, and the xx-axis is a center manifold. For each c∈ℝc \in \R, the curve y=ce1/xy = c e^{1/x} for x<0x < 0 together with y=0y = 0 for x≥0x \ge 0 is also a center manifold. On it y′=(−ce1/x/x2)⋅x2=−yy' = (-c e^{1/x} / x^2) \cdot x^2 = -y, and the curve is smooth and tangent to the xx-axis at 00. Non-example: the yy-axis is invariant and passes through the origin, but its tangent space is the stable direction, not EcE^c, so clause 1 fails. When Df(x0)Df(x_0) has no eigenvalue with positive real part, the stability of x0x_0 is decided by the reduced equation on a center manifold.

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To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Differential equation, concept, no evidence levelDifferential equationconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptInvariant set, concept, no evidence levelInvariant setconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionPartial derivative, concept, no evidence levelPartial derivativeconceptAutonomous system, definition, no evidence levelAutonomous systemdefinitionJacobian matrix, concept, no evidence levelJacobian matrixconceptEquilibrium point, definition, no evidence levelEquilibrium pointdefinitionCenter manifold, definition, no evidence levelCenter manifolddefinition
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. Euclidean space object
  3. Function concept
  4. Interval concept
  5. Invariant set concept
  6. Ordinary differential equation definition
  7. Total derivative concept
  8. Function of several variables concept
  9. Solution of an ordinary differential equation definition
  10. Partial derivative concept
  11. Autonomous system definition
  12. Jacobian matrix concept
  13. Equilibrium point definition
  14. Center manifold definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Center_manifold
About this page
TagF35WJP8VHG
URIm2://F35WJP8VHG
Kinddefinition
WritingW0 stub
MSC34C45, 37D10
DefinesCenter manifold
UsesAutonomous system, Equilibrium point, Jacobian matrix, Invariant set
Changed2 November 2026

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