Math 2.0

Partial differential equations Definition

Global attractor

Builds on Metric spaceCompact setInvariant set

Definition

Let (X,d)(X, d) be a metric space and let (S(t))t≥0(S(t))_{t \ge 0} be a semigroup of continuous maps S(t) ⁣:X→XS(t) \colon X \to X, with S(0)=id⁡S(0) = \id and S(t+s)=S(t)S(s)S(t + s) = S(t) S(s). For subsets A,B⊆XA, B \subseteq X, let dist⁡(B,A)=sup⁡b∈Binf⁡a∈Ad(a,b)\operatorname{dist}(B, A) = \sup_{b \in B} \inf_{a \in A} d(a, b). A global attractor of the semigroup is a set 𝒜⊆X\mathcal{A} \subseteq X such that:

  1. 𝒜\mathcal{A} is compact;
  2. 𝒜\mathcal{A} is invariant: S(t)𝒜=𝒜S(t) \mathcal{A} = \mathcal{A} for every t≥0t \ge 0;
  3. 𝒜\mathcal{A} attracts bounded sets: dist⁡(S(t)B,𝒜)→0\operatorname{dist}(S(t) B, \mathcal{A}) \to 0 as t→∞t \to \infty for every bounded set B⊆XB \subseteq X.

Remarks

A global attractor, when it exists, is unique: it is the largest compact invariant set and the smallest closed set that attracts bounded sets.

Example: let X=ℝX = \R and let S(t)S(t) be the flow of u′=u−u3u' = u - u^3. Every solution with u(0)>0u(0) > 0 tends to 11, every solution with u(0)<0u(0) < 0 tends to −1-1, and 00 is fixed. The set 𝒜=[−1,1]\mathcal{A} = [-1, 1] is compact, invariant since the flow maps [−1,1][-1, 1] onto itself, and attracts every bounded interval.

Non-example: for the same flow, the set {−1,1}\set{-1, 1} is compact and invariant, but it does not attract the bounded set {0}\set{0}, since S(t)0=0S(t) 0 = 0 for all tt. So clause 3 fails.

A standard sufficient condition is that the semigroup has a bounded absorbing set and is asymptotically compact.

Reading path

To understand this page from the ground up, read these 33 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionInvariant set, concept, no evidence levelInvariant setconceptMetric space, concept, no evidence levelMetric spaceconceptProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionSubspace topology, definition, no evidence levelSubspace topologydefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionOpen cover, definition, no evidence levelOpen coverdefinitionCompact space, definition, no evidence levelCompact spacedefinitionCompact set, definition, no evidence levelCompact setdefinitionGlobal attractor, definition, no evidence levelGlobal attractordefinition
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. Element definition
  2. Invariant set concept
  3. Metric space concept
  4. Proposition concept
  5. Real numbers object
  6. Set definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Predicate definition
  11. Empty set object
  12. Family of sets definition
  13. Ordered pair definition
  14. Intersection definition
  15. Union definition
  16. Existential quantifier definition
  17. Universal quantifier definition
  18. Cartesian product definition
  19. Subset definition
  20. Union of a family of sets definition
  21. Uniqueness quantifier definition
  22. Binary relation definition
  23. Topological space definition
  24. Function definition
  25. Subspace topology definition
  26. Metric space definition
  27. Open ball definition
  28. Metric topology definition
  29. Open set definition
  30. Open cover definition
  31. Compact space definition
  32. Compact set definition
  33. Global attractor definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Attractor
About this page
Tag681NY7506V
URIm2://681NY7506V
Kinddefinition
WritingW0 stub
MSC35B41, 37L25, 37L30
DefinesGlobal attractor
UsesMetric space, Compact set, Invariant set
Changed2 November 2026

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