A topological method for the detection of normally hyperbolic - PowerPoint PPT Presentation

A topological method for the detection of normally hyperbolic invariant manifolds Maciej Capi´ nski AGH University of Science and Technology, Krak´ ow Joint work with Piotr Zgliczy´ nski Jagiellonian University, Krak´ ow S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 1 / 18

Plan of the presentation Statement of the problem Normally hyperbolic invariant manifold theorem Covering relations and cone conditions Existence of the normally hyperbolic invariant manifold Foliation of W u and W s Verification of conditions Example S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 2 / 18

Statement of the problem f : Λ × B u × B s → Λ × R u × R s ( Λ = S 1 ) Λ is compact manifold without a boundary y ? ? x Do we have an invariant manifold in Λ × B u × B s ? S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 3 / 18

Normally hyperbolic invariant manifold theorem D = D = y y x x f : D → Λ × R 2 f ε = f + ε g we start with the region D and devise conditions which ensure the existence of the manifold the conditions are verifiable with rigorous numerics S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 4 / 18

Normally hyperbolic invariant manifold theorem D = D = y y x x f : D → Λ × R 2 we start with the region D and devise conditions which ensure the existence of the manifold the conditions are verifiable with rigorous numerics S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 4 / 18

Local maps Topological conditions (covering relations) D = f } } f y k i x x { V j } and { U i } are coverings of Λ � ⊂ U k × R u × R s � V j × B u × B s f k i S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 5 / 18

Cones y f k i x x In local coordinates we define Q ( θ , x , y ) = � x � 2 − � y � 2 − � θ � 2 Horizontal cone Q ≥ 0: For each point q ∈ D we have local coordinates which contain cones starting from q . Q = a and Q = b for 0 < a < b : y q x S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 6 / 18

Cone conditions y f k i x x m > 1. If Q ( x 1 − x 2 ) ≥ 0 then Q ( f ki ( x 1 ) − f ki ( x 2 )) > mQ ( x 1 − x 2 ) Horizontal cone Q ≥ 0: For each point q ∈ D we have local coordinates which contain cones starting from q . Q = a and Q = b for 0 < a < b : y q x S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 7 / 18

Horizontal discs y f k i x x If Q ( x 1 − x 2 ) ≥ 0 then Q ( f ki ( x 1 ) − f ki ( x 2 )) > mQ ( x 1 − x 2 ) A horizontal disc: A horizontal disc which b : B u → V j × B u × B s satisfies cone conditions: y y x x Lemma An image of a horizontal disc which satisfies cone conditions is a horizontal disc which satisfies cone conditions. S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 8 / 18

Horizontal discs y f k i x x If Q ( x 1 − x 2 ) ≥ 0 then Q ( f ki ( x 1 ) − f ki ( x 2 )) > mQ ( x 1 − x 2 ) Lemma An image of a horizontal disc which satisfies cone conditions is a horizontal disc which satisfies cone conditions. Proof. f k i y y x 0 x x S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 9 / 18

Horizontal discs y f k i x x If Q ( x 1 − x 2 ) ≥ 0 then Q ( f ki ( x 1 ) − f ki ( x 2 )) > mQ ( x 1 − x 2 ) Lemma An image of a horizontal disc which satisfies cone conditions is a horizontal disc which satisfies cone conditions. Proof. f k i y y x 0 x x S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 9 / 18

Horizontal discs y f k i x x If Q ( x 1 − x 2 ) ≥ 0 then Q ( f ki ( x 1 ) − f ki ( x 2 )) > mQ ( x 1 − x 2 ) Lemma An image of a horizontal disc which satisfies cone conditions is a horizontal disc which satisfies cone conditions. Proof. f k i y y x x � S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 9 / 18

Forward iterations y f k i x x If Q ( x 1 − x 2 ) ≥ 0 Q ( f ki ( x 1 ) − f ki ( x 2 )) > mQ ( x 1 − x 2 ) then Lemma For any θ 0 ∈ Λ we have a vertical disc of points in { θ 0 } × B u × B s which stay inside of Λ × B u × B s . Proof. y . . . 0 x S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 10 / 18

Forward iterations y f k i x x If Q ( x 1 − x 2 ) ≥ 0 Q ( f ki ( x 1 ) − f ki ( x 2 )) > mQ ( x 1 − x 2 ) then Lemma For any θ 0 ∈ Λ we have a vertical disc of points in { θ 0 } × B u × B s which stay inside of Λ × B u × B s . Proof. y . . . 0 x S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 10 / 18

Forward iterations y f k i x x If Q ( x 1 − x 2 ) ≥ 0 Q ( f ki ( x 1 ) − f ki ( x 2 )) > mQ ( x 1 − x 2 ) then Lemma For any θ 0 ∈ Λ we have a vertical disc of points in { θ 0 } × B u × B s which stay inside of Λ × B u × B s . Proof. ?! y . . . 0 x S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 10 / 18

Forward iterations y f k i x x If Q ( x 1 − x 2 ) ≥ 0 Q ( f ki ( x 1 ) − f ki ( x 2 )) > mQ ( x 1 − x 2 ) then Lemma For any θ 0 ∈ Λ we have a vertical disc of points in { θ 0 } × B u × B s which stay inside of Λ × B u × B s . Proof. y 0 x S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 10 / 18

Forward iterations y f k i x x If Q ( x 1 − x 2 ) ≥ 0 Q ( f ki ( x 1 ) − f ki ( x 2 )) > mQ ( x 1 − x 2 ) then Lemma For any θ 0 ∈ Λ we have a vertical disc of points in { θ 0 } × B u × B s which stay inside of Λ × B u × B s . Proof. y 0 x S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 10 / 18

Forward iterations y f k i x x If Q ( x 1 − x 2 ) ≥ 0 Q ( f ki ( x 1 ) − f ki ( x 2 )) > mQ ( x 1 − x 2 ) then Lemma For any θ 0 ∈ Λ we have a vertical disc of points in { θ 0 } × B u × B s which stay inside of Λ × B u × B s . Proof. y 0 x � S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 10 / 18

Main Result y f k i x x If Q ( x 1 − x 2 ) ≥ 0 then Q ( f ki ( x 1 ) − f ki ( x 2 )) > mQ ( x 1 − x 2 ) Theorem If f and f − 1 satisfy the the topological and cone conditions then there exists a C 0 map χ : Λ → Λ × B u × B s such that χ ( Λ ) = inv ( f , Λ × B u × B s ) and C 0 stable and unstable manifolds W s , W u . Proof a vertical disc of forward invariant points: y x 0 S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 11 / 18

Main Result y f k i x x If Q ( x 1 − x 2 ) ≥ 0 then Q ( f ki ( x 1 ) − f ki ( x 2 )) > mQ ( x 1 − x 2 ) Theorem If f and f − 1 satisfy the the topological and cone conditions then there exists a C 0 map χ : Λ → Λ × B u × B s such that χ ( Λ ) = inv ( f , Λ × B u × B s ) and C 0 stable and unstable manifolds W s , W u . Proof a horizontal disc of backward invariant points: y x 0 S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 11 / 18

Main Result y f k i x x If Q ( x 1 − x 2 ) ≥ 0 then Q ( f ki ( x 1 ) − f ki ( x 2 )) > mQ ( x 1 − x 2 ) Theorem If f and f − 1 satisfy the the topological and cone conditions then there exists a C 0 map χ : Λ → Λ × B u × B s such that χ ( Λ ) = inv ( f , Λ × B u × B s ) and C 0 stable and unstable manifolds W s , W u . Proof gives χ ( θ 0 ) : = q y q � x 0 S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 11 / 18

Foliation of W s A very simple example y y ′ = − 2 y θ ′ = − θ y ( t ) = y 0 e − 2 t θ ( t ) = θ 0 e − t Vertical cone: V ≥ 0 V ( θ , x , y ) = −� θ � 2 − � x � 2 + � y � 2 S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 12 / 18

Foliation of W s A very simple example y ′ = − 2 y θ ′ = − θ y ( t ) = y 0 e − 2 t θ ( t ) = θ 0 e − t Vertical cone: V ≥ 0 V ( θ , x , y ) = −� θ � 2 − � x � 2 + � y � 2 S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 12 / 18

Foliation of W s y Foliation conditions 0 < β < λ , q 1 � = q 2 1 If V ( q 1 − q 2 ) ≥ 0 then � π y ( f ( q 1 ) − f ( q 2 )) � < β � π y ( q 1 − q 2 ) � 2 If V ( q 1 − q 2 ) < 0 then V ( f ( q 1 ) − f ( q 2 )) < λ 2 V ( q 1 − q 2 ) V ( θ , x , y ) = −� θ � 2 − � x � 2 + � y � 2 S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 13 / 18

Foliation of W s y Foliation conditions V ≥ 0, V = c < 0, V = λ 2 c 0 < β < λ , q 1 � = q 2 1 If V ( q 1 − q 2 ) ≥ 0 then � π y ( f ( q 1 ) − f ( q 2 )) � < β � π y ( q 1 − q 2 ) � 2 If V ( q 1 − q 2 ) < 0 then V ( f ( q 1 ) − f ( q 2 )) < λ 2 V ( q 1 − q 2 ) V ( θ , x , y ) = −� θ � 2 − � x � 2 + � y � 2 S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 13 / 18