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´

• w

Joint work with

Piotr Zgliczy´ nski

Jagiellonian University, Krak´

• w

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 × Bu × Bs → Λ × Ru × Rs Λ is compact manifold without a boundary (Λ = S1)

y x

? ?

Do we have an invariant manifold in Λ × Bu × Bs?

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 3 / 18

Normally hyperbolic invariant manifold theorem

y x y x

D = D =

f : D → Λ × R2 fε = f + εg 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

y x y x

D = D =

f : D → Λ × R2 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

x y k i x

{Vj} and {Ui} are coverings of Λ fk i

• Vj × Bu × Bs

⊂ Uk × Ru × Rs

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 5 / 18

Cones

In local coordinates we define Q(θ, x, y) = x2 − y2 − θ2 Horizontal cone Q ≥ 0: Q = a and Q = b for 0 < a < b: For each point q ∈ D we have local coordinates which contain cones starting from q.

x y q

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 6 / 18

f

x y k i x

Cone conditions

m > 1. If Q(x1 − x2) ≥ 0 then Q(fki(x1) − fki(x2)) > mQ(x1 − x2) Horizontal cone Q ≥ 0: Q = a and Q = b for 0 < a < b: For each point q ∈ D we have local coordinates which contain cones starting from q.

x y q

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 7 / 18

f

x y k i x

Horizontal discs

If Q(x1 − x2) ≥ 0 then Q(fki(x1) − fki(x2)) > mQ(x1 − x2) A horizontal disc: b : Bu → Vj × Bu × Bs

x y

A horizontal disc which satisfies cone conditions:

x y

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

f

x y k i x

Horizontal discs

If Q(x1 − x2) ≥ 0 then Q(fki(x1) − fki(x2)) > mQ(x1 − x2)

Lemma

An image of a horizontal disc which satisfies cone conditions is a horizontal disc which satisfies cone conditions. Proof.

x y x y x0

fk i

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 9 / 18

f

x y k i x

Horizontal discs

If Q(x1 − x2) ≥ 0 then Q(fki(x1) − fki(x2)) > mQ(x1 − x2)

Lemma

An image of a horizontal disc which satisfies cone conditions is a horizontal disc which satisfies cone conditions. Proof.

x y x y x0

fk i

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 9 / 18

f

x y k i x

Horizontal discs

If Q(x1 − x2) ≥ 0 then Q(fki(x1) − fki(x2)) > mQ(x1 − x2)

Lemma

An image of a horizontal disc which satisfies cone conditions is a horizontal disc which satisfies cone conditions. Proof.

x y x y

fk i

• S.I.M.S. Workshop

Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 9 / 18

f

x y k i x

Forward iterations

If Q(x1 − x2) ≥ 0 then Q(fki(x1) − fki(x2)) > mQ(x1 − x2)

Lemma

For any θ0 ∈ Λ we have a vertical disc of points in {θ0} × Bu × Bs which stay inside of Λ × Bu × Bs. Proof.

x y

. . .

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 10 / 18

f

x y k i x

Forward iterations

If Q(x1 − x2) ≥ 0 then Q(fki(x1) − fki(x2)) > mQ(x1 − x2)

Lemma

For any θ0 ∈ Λ we have a vertical disc of points in {θ0} × Bu × Bs which stay inside of Λ × Bu × Bs. Proof.

x y

. . .

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 10 / 18

f

x y k i x

Forward iterations

If Q(x1 − x2) ≥ 0 then Q(fki(x1) − fki(x2)) > mQ(x1 − x2)

Lemma

For any θ0 ∈ Λ we have a vertical disc of points in {θ0} × Bu × Bs which stay inside of Λ × Bu × Bs. Proof.

x y

. . .

?!

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 10 / 18

f

x y k i x

Forward iterations

If Q(x1 − x2) ≥ 0 then Q(fki(x1) − fki(x2)) > mQ(x1 − x2)

Lemma

For any θ0 ∈ Λ we have a vertical disc of points in {θ0} × Bu × Bs which stay inside of Λ × Bu × Bs. Proof.

x y

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 10 / 18

f

x y k i x

Forward iterations

If Q(x1 − x2) ≥ 0 then Q(fki(x1) − fki(x2)) > mQ(x1 − x2)

Lemma

For any θ0 ∈ Λ we have a vertical disc of points in {θ0} × Bu × Bs which stay inside of Λ × Bu × Bs. Proof.

x y

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 10 / 18

f

x y k i x

Forward iterations

If Q(x1 − x2) ≥ 0 then Q(fki(x1) − fki(x2)) > mQ(x1 − x2)

Lemma

For any θ0 ∈ Λ we have a vertical disc of points in {θ0} × Bu × Bs which stay inside of Λ × Bu × Bs. Proof.

x y

• S.I.M.S. Workshop

Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 10 / 18

f

x y k i x

Main Result

If Q(x1 − x2) ≥ 0 then Q(fki(x1) − fki(x2)) > mQ(x1 − x2)

Theorem

If f and f −1 satisfy the the topological and cone conditions then there exists a C 0 map χ : Λ → Λ × Bu × Bs such that χ(Λ) = inv(f , Λ × Bu × Bs) and C 0 stable and unstable manifolds W s, W u. Proof a vertical disc of forward invariant points:

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 11 / 18 x y

f

x y k i x

Main Result

If Q(x1 − x2) ≥ 0 then Q(fki(x1) − fki(x2)) > mQ(x1 − x2)

Theorem

If f and f −1 satisfy the the topological and cone conditions then there exists a C 0 map χ : Λ → Λ × Bu × Bs such that χ(Λ) = inv(f , Λ × Bu × Bs) and C 0 stable and unstable manifolds W s, W u. Proof a horizontal disc of backward invariant points:

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 11 / 18 x y

f

x y k i x

Main Result

If Q(x1 − x2) ≥ 0 then Q(fki(x1) − fki(x2)) > mQ(x1 − x2)

Theorem

If f and f −1 satisfy the the topological and cone conditions then there exists a C 0 map χ : Λ → Λ × Bu × Bs such that χ(Λ) = inv(f , Λ × Bu × Bs) and C 0 stable and unstable manifolds W s, W u. Proof gives χ(θ0) := q

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 11 / 18

• x

y

q

f

x y k i x

Foliation of W s

A very simple example

y ′ = −2y θ′ = −θ y(t) = y0e−2t θ(t) = θ0e−t

y

Vertical cone: V ≥ 0 V (θ, x, y) = −θ2 − x2 + y2

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 12 / 18

Foliation of W s

A very simple example

y ′ = −2y θ′ = −θ y(t) = y0e−2t θ(t) = θ0e−t Vertical cone: V ≥ 0 V (θ, x, y) = −θ2 − x2 + y2

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 12 / 18

Foliation of W s

Foliation conditions

0 < β < λ, q1 = q2

1 If V (q1 − q2) ≥ 0 then πy(f (q1) − f (q2)) < βπy(q1 − q2) 2 If V (q1 − q2) < 0 then V (f (q1) − f (q2)) < λ2V (q1 − q2)

V (θ, x, y) = −θ2 − x2 + y2

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 13 / 18

y

Foliation of W s

Foliation conditions

0 < β < λ, q1 = q2

1 If V (q1 − q2) ≥ 0 then πy(f (q1) − f (q2)) < βπy(q1 − q2) 2 If V (q1 − q2) < 0 then V (f (q1) − f (q2)) < λ2V (q1 − q2)

V (θ, x, y) = −θ2 − x2 + y2

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 13 / 18

y

V ≥ 0, V = c < 0, V = λ2c

Foliation of W s

Foliation conditions

0 < β < λ, q1 = q2

1 If V (q1 − q2) ≥ 0 then πy(f (q1) − f (q2)) < βπy(q1 − q2) 2 If V (q1 − q2) < 0 then V (f (q1) − f (q2)) < λ2V (q1 − q2)

V (θ, x, y) = −θ2 − x2 + y2

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 13 / 18

V ≥ 0, V = c < 0, V = λ2c

Foliation of W s

Foliation conditions

0 < β < λ, q1 = q2

1 If V (q1 − q2) ≥ 0 then πy(f (q1) − f (q2)) < βπy(q1 − q2) 2 If V (q1 − q2) < 0 then V (f (q1) − f (q2)) < λ2V (q1 − q2)

V (θ, x, y) = −θ2 − x2 + y2

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 13 / 18

V ≥ 0, V = c < 0, V = λ2c

Foliation of W s

Foliation conditions

0 < β < λ, q1 = q2

1 If V (q1 − q2) ≥ 0 then πy(f (q1) − f (q2)) < βπy(q1 − q2) 2 If V (q1 − q2) < 0 then V (f (q1) − f (q2)) < λ2V (q1 − q2)

V (θ, x, y) = −θ2 − x2 + y2

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 13 / 18

V ≥ 0, V = c < 0, V = λ2c

Foliation of W s

0 < β < λ, q1 = q2

1 If V (q1 − q2) ≥ 0 then πy(f (q1) − f (q2)) < βπy(q1 − q2) 2 If V (q1 − q2) < 0 then V (f (q1) − f (q2)) < λ2V (q1 − q2)

Theorem

For each q ∈ χ(Λ) there exists a vertical disc b = W s

q i.e.

f n(b(y)) − f n(q) < C βnb(y) − q Proof.

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 14 / 18

x y

. . .

y0

f (q)

k

q

y

V ≥ 0, V = c < 0, V = λ2c

Foliation of W s

0 < β < λ, q1 = q2

1 If V (q1 − q2) ≥ 0 then πy(f (q1) − f (q2)) < βπy(q1 − q2) 2 If V (q1 − q2) < 0 then V (f (q1) − f (q2)) < λ2V (q1 − q2)

Theorem

For each q ∈ χ(Λ) there exists a vertical disc b = W s

q i.e.

f n(b(y)) − f n(q) < C βnb(y) − q Proof.

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 14 / 18

x y

. . .

y0

f (q)

k

q

y

V ≥ 0, V = c < 0, V = λ2c

Foliation of W s

0 < β < λ, q1 = q2

1 If V (q1 − q2) ≥ 0 then πy(f (q1) − f (q2)) < βπy(q1 − q2) 2 If V (q1 − q2) < 0 then V (f (q1) − f (q2)) < λ2V (q1 − q2)

Theorem

For each q ∈ χ(Λ) there exists a vertical disc b = W s

q i.e.

f n(b(y)) − f n(q) < C βnb(y) − q Proof.

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 14 / 18

x y

. . .

y0

f (q)

k

q pk

y

V ≥ 0, V = c < 0, V = λ2c

Foliation of W s

0 < β < λ, q1 = q2

1 If V (q1 − q2) ≥ 0 then πy(f (q1) − f (q2)) < βπy(q1 − q2) 2 If V (q1 − q2) < 0 then V (f (q1) − f (q2)) < λ2V (q1 − q2)

Theorem

For each q ∈ χ(Λ) there exists a vertical disc b = W s

q i.e.

f n(b(y)) − f n(q) < C βnb(y) − q Proof.

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 14 / 18

pk → b(y0), f n(b(y0)) − f n(q) < C βnb(y0) − q

x y y0

q

y

V ≥ 0, V = c < 0, V = λ2c

Foliation of W s

0 < β < λ, q1 = q2

1 If V (q1 − q2) ≥ 0 then πy(f (q1) − f (q2)) < βπy(q1 − q2) 2 If V (q1 − q2) < 0 then V (f (q1) − f (q2)) < λ2V (q1 − q2)

Theorem

For each q ∈ χ(Λ) there exists a vertical disc b = W s

q i.e.

f n(b(y)) − f n(q) < C βnb(y) − q Proof.

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 14 / 18

pk → b(y0), f n(b(y0)) − f n(q) < C βnb(y0) − q f n(p1) − f n(p2) < βnc1

x y y0

q p1 p2

y

V ≥ 0, V = c < 0, V = λ2c

Foliation of W s

0 < β < λ, q1 = q2

1 If V (q1 − q2) ≥ 0 then πy(f (q1) − f (q2)) < βπy(q1 − q2) 2 If V (q1 − q2) < 0 then V (f (q1) − f (q2)) < λ2V (q1 − q2)

Theorem

For each q ∈ χ(Λ) there exists a vertical disc b = W s

q i.e.

f n(b(y)) − f n(q) < C βnb(y) − q Proof.

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 14 / 18

pk → b(y0), f n(b(y0)) − f n(q) < C βnb(y0) − q f n(p1) − f n(p2) < βnc1 f n(p1) − f n(p2) > λnc2 !!

x y y0

q p1 p2

y

V ≥ 0, V = c < 0, V = λ2c

Foliation of W s

0 < β < λ, q1 = q2

1 If V (q1 − q2) ≥ 0 then πy(f (q1) − f (q2)) < βπy(q1 − q2) 2 If V (q1 − q2) < 0 then V (f (q1) − f (q2)) < λ2V (q1 − q2)

Theorem

For each q ∈ χ(Λ) there exists a vertical disc b = W s

q i.e.

f n(b(y)) − f n(q) < C βnb(y) − q Proof.

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 14 / 18

pk → b(y0), f n(b(y0)) − f n(q) < C βnb(y0) − q f n(p1) − f n(p2) < βnc1 f n(p1) − f n(p2) > λnc2 !!

• x

y

q

y

V ≥ 0, V = c < 0, V = λ2c

What can we do so far?

y x

? ?

We have a normally hyperbolic invariant manifold in Λ × Bu × Bs We have it’s stable and unstable manifolds W s and W u We have foliations of W s and W u Question: How can we verify our assumptions in practice?

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 15 / 18

Verification of conditions

If Q(x1 − x2) ≥ 0 then Q(fki(x1) − fki(x2)) > mQ(x1 − x2) We need [Df (Vj)] ← →     

• ∂f1

∂θ

• ≤ C

ε ε ε

• ∂f2

∂x

• ≥ α

ε ε ε

• ∂f3

∂y

• ≤ β

     where β < C < α with β < 1 < α and ε appropriately small.

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 16 / 18

f

x y k i x

y

Example of applications

Rotating H´ enon map

¯ θ = θ + ω (mod 1), ¯ x = 1 + y − ax2 + ε cos(2πθ), ¯ y = bx For a = 0.68, b = 0.1 and ε ≤ 1

2

Λ ⊂ Uε = T1 × [x0 − 1.1ε, x0 + 1.1ε] × [y0 − 0.12ε, y0 + 0.12ε], where x0 = −(1 − b) −

• (1 − b)2 + 4a

2a ≈ −2.043 3, y0 = bx0 ≈ −0.204 33.

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 17 / 18

y x (x ,y )

Example of applications

Rotating H´ enon map

¯ θ = θ + ω (mod 1), ¯ x = 1 + y − ax2 + ε cos(2πθ), ¯ y = bx For a = 0.68, b = 0.1 and ε ≤ 1

2

Λ ⊂ Uε = T1 × [x0 − 1.1ε, x0 + 1.1ε] × [y0 − 0.12ε, y0 + 0.12ε], where x0 = −(1 − b) −

• (1 − b)2 + 4a

2a ≈ −2.043 3, y0 = bx0 ≈ −0.204 33.

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 17 / 18

y x (x ,y )

Closing remarks

The method works in a more general setting. We do not need an invariant manifold to start with. If we start with an invariant manifold then we can estimate the size of the perturbation for which it survives. We only know that the invariant manifold, W u, W s, and foliations are C 0. The method still waits to be tested on a more challenging example.

S.I.M.S. Workshop Normally Hyperbolic Manifolds Barcelona, 1 Dec. 2008 18 / 18