Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant...

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Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH University of Science and Technology, Kraków JISD2012 Geometric methods for manifolds I. 28 May 2012 1 / 27

Transcript of Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant...

Page 1: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Geometric methods for invariant manifolds in dynamicalsystems I.

Fixed points and periodic orbits

Maciej Capiński

AGH University of Science and Technology, Kraków

JISD2012 Geometric methods for manifolds I. 28 May 2012 1 / 27

Page 2: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Geometric methods for invariant manifolds in dynamicalsystems I.

Fixed points and periodic orbits

Maciej Capiński

AGH University of Science and Technology, Kraków

M. Gidea, T. Kapela, C. Simó, P. Roldan, D. Wilczak, P. Zgliczyński

JISD2012 Geometric methods for manifolds I. 28 May 2012 1 / 27

Page 3: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Plan of the lecture

Motivation

Examples of methodology that we shall use

Brouwer theorem

Interval Newton method

Covering relations

Example of application

JISD2012 Geometric methods for manifolds I. 28 May 2012 2 / 27

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Motivation

ODEs PDEsx = f (x)

Fixed point f (p) = 0

f : R2 → R2

JISD2012 Geometric methods for manifolds I. 28 May 2012 3 / 27

Page 5: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Motivation

ODEs PDEsx = f (x)

Periodic orbit

f : R2 → R2

JISD2012 Geometric methods for manifolds I. 28 May 2012 3 / 27

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Motivation

ODEs PDEsx = f (x)

Periodic orbit

f : R2 → R2

JISD2012 Geometric methods for manifolds I. 28 May 2012 3 / 27

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Motivation

ODEs PDEsx = f (x)

Stable, unstable manifolds

f : R2 → R2

JISD2012 Geometric methods for manifolds I. 28 May 2012 3 / 27

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Motivation

ODEs PDEsx = f (x)

Stable, unstable manifolds

f : R2 → R2

JISD2012 Geometric methods for manifolds I. 28 May 2012 3 / 27

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Motivation

ODEs PDEsx = f (x)

Normally hyperbolic manifolds

f : R2 × S1 → R2 × S1

JISD2012 Geometric methods for manifolds I. 28 May 2012 3 / 27

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Motivation

ODEs PDEsx = f (x)

Normally hyperbolic manifolds

f : R3 → R3

JISD2012 Geometric methods for manifolds I. 28 May 2012 3 / 27

Page 11: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Motivation

ODEs PDEsx = f (x)

Normally hyperbolic manifolds

f : Rn → Rn

JISD2012 Geometric methods for manifolds I. 28 May 2012 3 / 27

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Motivation

ODEs PDEsx = f (x)

Normally hyperbolic manifolds

f : Rn → Rn

ut = Lu +N

u(t, x) =∑k

ak(t)eikπx

Take a = (ai , . . . , ai+n)

a = f (a) + R

ak < 0 for k /∈ {i , . . . , i + n}.

JISD2012 Geometric methods for manifolds I. 28 May 2012 3 / 27

Page 13: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Motivation

ODEs PDEsx = f (x)

Normally hyperbolic manifolds

f : Rn → Rn

ut = Lu +N

u(t, x) =∑k

ak(t)eikπx

Take a = (ai , . . . , ai+n)

a = f (a) + R

ak < 0 for k /∈ {i , . . . , i + n}.

JISD2012 Geometric methods for manifolds I. 28 May 2012 3 / 27

Page 14: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Motivation

ODEs PDEsx = f (x)

Normally hyperbolic manifolds

f : Rn → Rn

ut = Lu +N

u(t, x) =∑k

ak(t)eikπx

Take a = (ai , . . . , ai+n)

a = f (a) + R

If ak < 0 for k /∈ {i , . . . , i + n}

JISD2012 Geometric methods for manifolds I. 28 May 2012 3 / 27

Page 15: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Motivation

ODEs PDEsx = f (x)

Normally hyperbolic manifolds

f : Rn → Rn

ut = Lu +N

u(t, x) =∑k

ak(t)eikπx

Take a = (ai , . . . , ai+n)

a = f (a) + R

If ak < 0 for k /∈ {i , . . . , i + n}

JISD2012 Geometric methods for manifolds I. 28 May 2012 3 / 27

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Kinds of tools that we shall useBolzano theorem

f : R→ R f (x) ?= 0

a b

f (a) < 0 f (b) > 0

There exists an x∗ in (a, b) such that

f (x∗) = 0

JISD2012 Geometric methods for manifolds I. 28 May 2012 4 / 27

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Kinds of tools that we shall useBolzano theorem - no need to be too accurate

f : R→ R f (x) ?= 0

a b

f (a) < 0 f (b) > 0

There exists an x∗ in (a, b) such that

f (x∗) = 0

JISD2012 Geometric methods for manifolds I. 28 May 2012 5 / 27

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Kinds of tools that we shall useBolzano theorem - some more information

f : R→ R f ′ : R→ R

a b

a b

f (a) < 0 f (b) > 0 f ′(x) > 0, x ∈ [a, b]

There exists a unique x∗ in (a, b) such that

f (x∗) = 0

JISD2012 Geometric methods for manifolds I. 28 May 2012 6 / 27

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Kinds of tools that we shall useInterval arithmetic

computations on intervals:

[1, 2] + [3, 4] = [4, 6]

[1, 2]− [3, 4] = [−3,−1]

[1, 2] ∗ [3, 4] = [3, 12]

[1, 2]/[3, 4] = [14,

23]

[1, 2][3,4] = [13, 24]

...

extends to higher dimensions

[1, 2]− [1, 2] = [−1, 1]

f : Rn → Rn

What can be computed:

[f (U)]

[Df (U)]

higher order derivatives

linear algebra; eg. [A−1]

CAPD library

JISD2012 Geometric methods for manifolds I. 28 May 2012 7 / 27

a b

Page 20: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Kinds of tools that we shall useInterval arithmetic

computations on intervals:

[1, 2] + [3, 4] = [4, 6]

[1, 2]− [3, 4] = [−3,−1]

[1, 2] ∗ [3, 4] = [3, 12]

[1, 2]/[3, 4] = [14,

23]

[1, 2][3,4] = [13, 24]

...

extends to higher dimensions

[1, 2]− [1, 2] = [−1, 1]

f : Rn → Rn

What can be computed:

[f (U)]

[Df (U)]

higher order derivatives

linear algebra; eg. [A−1]

CAPD library

JISD2012 Geometric methods for manifolds I. 28 May 2012 7 / 27

a b

Page 21: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Kinds of tools that we shall useInterval arithmetic

computations on intervals:

[1, 2] + [3, 4] = [4, 6]

[1, 2]− [3, 4] = [−3,−1]

[1, 2] ∗ [3, 4] = [3, 12]

[1, 2]/[3, 4] = [14,

23]

[1, 2][3,4] = [13, 24]

...

extends to higher dimensions

[1, 2]− [1, 2] = [−1, 1]

f : Rn → Rn

What can be computed:

[f (U)]

[Df (U)]

higher order derivatives

linear algebra; eg. [A−1]

CAPD library

JISD2012 Geometric methods for manifolds I. 28 May 2012 7 / 27

a b

Page 22: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Kinds of tools that we shall useBrouwer theorem

x = f (x) f (x∗) ?= 0

Theorem (Brouwer theorem)

If F : B → B is continuous, thenthere exists a q ∈ B

F (q) = q

F

F(B)

JISD2012 Geometric methods for manifolds I. 28 May 2012 8 / 27

Page 23: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Interval Newton methodF : Rn → Rn, C 1

B = Πni=1[ai , bi ]

x0 ∈ B

Theorem (interval Newton)

Ifx0 − [DF (B)]−1F (x0) ⊂ B

Then ∃!x∗ ∈ B such that

F (x∗) = 0

(intuition)Newton-Raphson:

F : R→ R

xn+1 = xn−F (xn)F ′(xn)

Proof. chalk.

JISD2012 Geometric methods for manifolds I. 28 May 2012 9 / 27

F

F(B)

Theorem (Brouwer)If F : B → B is continuous,then there exists a q ∈ B suchthat F (q) = q.

Page 24: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Interval Newton methodF : Rn → Rn, C 1

B = Πni=1[ai , bi ]

x0 ∈ B

Theorem (interval Newton)

Ifx0 − [DF (B)]−1F (x0) ⊂ B

Then ∃!x∗ ∈ B such that

F (x∗) = 0

(intuition)Newton-Raphson:

F : R→ R

xn+1 = xn−F (xn)F ′(xn)

Proof. chalk.

JISD2012 Geometric methods for manifolds I. 28 May 2012 9 / 27

F

F(B)

Theorem (Brouwer)If F : B → B is continuous,then there exists a q ∈ B suchthat F (q) = q.

Page 25: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Interval Newton methodF : Rn → Rn, C 1

B = Πni=1[ai , bi ]

x0 ∈ B

Theorem (interval Newton)

Ifx0 − [DF (B)]−1F (x0) ⊂ B

Then ∃!x∗ ∈ B such that

F (x∗) = 0

(intuition)Newton-Raphson:

F : R→ R

xn+1 = xn−F (xn)F ′(xn)

Proof. chalk.

JISD2012 Geometric methods for manifolds I. 28 May 2012 9 / 27

F

F(B)

Theorem (Brouwer)If F : B → B is continuous,then there exists a q ∈ B suchthat F (q) = q.

Page 26: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Interval Newton methodF : Rn → Rn, C 1

B = Πni=1[ai , bi ]

x0 ∈ B

Theorem (interval Newton)

Ifx0 − [DF (B)]−1F (x0) ⊂ B

Then ∃!x∗ ∈ B such that

F (x∗) = 0

(intuition)Newton-Raphson:

F : R→ R

xn+1 = xn−F (xn)F ′(xn)

Proof. chalk.

JISD2012 Geometric methods for manifolds I. 28 May 2012 9 / 27

F

F(B)

Theorem (Brouwer)If F : B → B is continuous,then there exists a q ∈ B suchthat F (q) = q.

Page 27: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Interval Newton methodF : Rn → Rn, C 1

B = Πni=1[ai , bi ]

x0 ∈ B

Theorem (interval Newton)

Ifx0 − [DF (B)]−1F (x0) ⊂ B

Then ∃!x∗ ∈ B such that

F (x∗) = 0

(intuition)Newton-Raphson:

F : R→ R

xn+1 = xn−F (xn)F ′(xn)

Proof. chalk.

JISD2012 Geometric methods for manifolds I. 28 May 2012 9 / 27

F

F(B)

Theorem (Brouwer)If F : B → B is continuous,then there exists a q ∈ B suchthat F (q) = q.

Page 28: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Interval Newton methodF : Rn → Rn, C 1

B = Πni=1[ai , bi ]

x0 ∈ B

Theorem (interval Newton)

Ifx0 − [DF (B)]−1F (x0) ⊂ B

Then ∃!x∗ ∈ B such that

F (x∗) = 0

(intuition)Newton-Raphson:

F : R→ R

xn+1 = xn−F (xn)F ′(xn)

Proof. chalk.

JISD2012 Geometric methods for manifolds I. 28 May 2012 9 / 27

F

F(B)

Theorem (Brouwer)If F : B → B is continuous,then there exists a q ∈ B suchthat F (q) = q.

Page 29: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Interval Newton methodExample - Henon map

h(x , y) = (1− ax2 + y , bx)

a = 1.4, b = 0.3

Fixed point:

h(x , y) = (x , y)

F (x , y) = (1− ax2 + y − x , bx − y) = 0

JISD2012 Geometric methods for manifolds I. 28 May 2012 10 / 27

Theorem (interval Newton)

x0 − [DF (B)]−1F (x0) ⊂ B.

Then ∃!x∗ ∈ B, F (x∗) = 0.

Page 30: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Interval Newton methodExample - Henon map

h(x , y) = (1− ax2 + y , bx)

a = 1.4, b = 0.3

Fixed point:

h(x , y) = (x , y)

F (x , y) = (1− ax2 + y − x , bx − y) = 0

JISD2012 Geometric methods for manifolds I. 28 May 2012 10 / 27

Theorem (interval Newton)

x0 − [DF (B)]−1F (x0) ⊂ B.

Then ∃!x∗ ∈ B, F (x∗) = 0.

Page 31: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Interval Newton methodExample - Henon map

h(x , y) = (1− ax2 + y , bx)

a = 1.4, b = 0.3

Fixed point:

h(x , y) = (x , y)

F (x , y) = (1− ax2 + y − x , bx − y) = 0

JISD2012 Geometric methods for manifolds I. 28 May 2012 10 / 27

Theorem (interval Newton)

x0 − [DF (B)]−1F (x0) ⊂ B.

Then ∃!x∗ ∈ B, F (x∗) = 0.

Page 32: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Interval Newton methodExample - Henon map

F (x , y) = (1− ax2 + y − x , bx − y) = 0

JISD2012 Geometric methods for manifolds I. 28 May 2012 11 / 27

Theorem (interval Newton)

x0 − [DF (B)]−1F (x0) ⊂ B.

Then ∃!x∗ ∈ B, F (x∗) = 0.

Page 33: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Interval Newton methodExample - Henon map

F (x , y) = (1− ax2 + y − x , bx − y) = 0

JISD2012 Geometric methods for manifolds I. 28 May 2012 11 / 27

Theorem (interval Newton)

x0 − [DF (B)]−1F (x0) ⊂ B.

Then ∃!x∗ ∈ B, F (x∗) = 0.

Page 34: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Interval Newton methodExample - Henon map

F (x , y) = (1− ax2 + y − x , bx − y) = 0

JISD2012 Geometric methods for manifolds I. 28 May 2012 11 / 27

Theorem (interval Newton)

x0 − [DF (B)]−1F (x0) ⊂ B.

Then ∃!x∗ ∈ B, F (x∗) = 0.

Page 35: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Interval Newton methodExample - Henon map

F (x , y) = (1− ax2 + y − x , bx − y) = 0

JISD2012 Geometric methods for manifolds I. 28 May 2012 11 / 27

Theorem (interval Newton)

x0 − [DF (B)]−1F (x0) ⊂ B.

Then ∃!x∗ ∈ B, F (x∗) = 0.

Page 36: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Interval Newton methodExample - Henon map

F (x , y) = (1− ax2 + y − x , bx − y) = 0

JISD2012 Geometric methods for manifolds I. 28 May 2012 11 / 27

Theorem (interval Newton)

x0 − [DF (B)]−1F (x0) ⊂ B.

Then ∃!x∗ ∈ B, F (x∗) = 0.

Page 37: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Interval Newton methodExample - Henon map

F (x , y) = (1− ax2 + y − x , bx − y) = 0

JISD2012 Geometric methods for manifolds I. 28 May 2012 11 / 27

Theorem (interval Newton)

x0 − [DF (B)]−1F (x0) ⊂ B.

Then ∃!x∗ ∈ B, F (x∗) = 0.

Page 38: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

From ODEs to maps

Theorem (Brouwer theorem)

If F : B → B is continuous, thenthere exists a q ∈ B

F (q) = q

F

F(B)

JISD2012 Geometric methods for manifolds I. 28 May 2012 12 / 27

Page 39: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

From ODEs to maps

Theorem (Brouwer theorem)

If F : B → B is continuous, thenthere exists a q ∈ B

F (q) = q

F

F(B)

x = f (x , t)

t

x

T

JISD2012 Geometric methods for manifolds I. 28 May 2012 12 / 27

Page 40: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

From ODEs to maps

Theorem (Brouwer theorem)

If F : B → B is continuous, thenthere exists a q ∈ B

F (q) = q

F

F(B)

x = f (x , t) P(q) = q

B

t

x

B

T

JISD2012 Geometric methods for manifolds I. 28 May 2012 12 / 27

Page 41: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

From ODEs to mapsPoincare map

x = f (x)

x(0) = x0

Flowφ(t, x0) = x(t)

Time T -shift map

P(x) = φ(T , x)

f : Rn → Rn

V ⊂ Rn

Poincare map:

P : V → V

P(x) = φ(τ (x), x)

V

JISD2012 Geometric methods for manifolds I. 28 May 2012 13 / 27

Page 42: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

From ODEs to mapsPoincare map

x = f (x)

x(0) = x0

Flowφ(t, x0) = x(t)

Time T -shift map

P(x) = φ(T , x)

f : Rn → Rn

V ⊂ Rn

Poincare map:

P : V → V

P(x) = φ(τ (x), x)

V

JISD2012 Geometric methods for manifolds I. 28 May 2012 13 / 27

Page 43: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

From ODEs to mapsPoincare map

x = f (x)

x(0) = x0

Flowφ(t, x0) = x(t)

Time T -shift map

P(x) = φ(T , x)

f : Rn → Rn

V ⊂ Rn

Poincare map:

P : V → V

P(x) = φ(τ (x), x)

V

JISD2012 Geometric methods for manifolds I. 28 May 2012 13 / 27

Page 44: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Periodic orbits of ODEsInterval Newton method

x = f (x)

P : V → VP(x) = φ(τ (x), x)

P(x∗) = x∗

V

x0

TheoremIf

x0 − [DF (B)]−1F (x0) ⊂ B

Then ∃!x∗ ∈ B such that

F (x∗) = 0

F (x) = P(x)− x

x0− [DP(B)− Id ]−1(P(x0)− x0) ⊂ B

JISD2012 Geometric methods for manifolds I. 28 May 2012 14 / 27

Page 45: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Periodic orbits of ODEsInterval Newton method

x = f (x)

P : V → VP(x) = φ(τ (x), x)

P(x∗) = x∗

V

x0B

TheoremIf

x0 − [DF (B)]−1F (x0) ⊂ B

Then ∃!x∗ ∈ B such that

F (x∗) = 0

F (x) = P(x)− x

x0− [DP(B)− Id ]−1(P(x0)− x0) ⊂ B

JISD2012 Geometric methods for manifolds I. 28 May 2012 14 / 27

Page 46: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Periodic orbits of ODEsInterval Newton method

x = f (x)

P : V → VP(x) = φ(τ (x), x)

P(x∗) = x∗

V

x0B

TheoremIf

x0 − [DF (B)]−1F (x0) ⊂ B

Then ∃!x∗ ∈ B such that

F (x∗) = 0

F (x) = P(x)− x

x0− [DP(B)− Id ]−1(P(x0)− x0) ⊂ B

JISD2012 Geometric methods for manifolds I. 28 May 2012 14 / 27

Page 47: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Topological coveringF : Rn → Rn

N = Bu ×Bs

N− = ∂Bu ×BsN+ = Bu × ∂Bs

Definition (covering)

N F⇒ NπuF (N−) ∩Bu = ∅πsF (N) ⊂ Bs∃q0 ∈ N s.t. F (q0) ∈ intN (∗)

s

u

N

N-

TheoremThere exists a point p ∈ N suchthat

F (p) = p

Proof. chalk.(*) stronger conditions needed: ∃h : [0, 1]× N → Rn , homotopy, such that (see [GZ] for details):

h0 = F , hλ(N−) ∩ N = ∅, hλ(N) ∩ N+ = ∅ h1 = (A, 0), where A is a matrix s.t. Bu ⊂ ABu

JISD2012 Geometric methods for manifolds I. 28 May 2012 15 / 27

Page 48: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Topological coveringF : Rn → Rn

N = Bu ×Bs

N− = ∂Bu ×BsN+ = Bu × ∂Bs

Definition (covering)

N F⇒ NπuF (N−) ∩Bu = ∅πsF (N) ⊂ Bs∃q0 ∈ N s.t. F (q0) ∈ intN (∗)

s

u

N

N F⇒ N

TheoremThere exists a point p ∈ N suchthat

F (p) = p

Proof. chalk.(*) stronger conditions needed: ∃h : [0, 1]× N → Rn , homotopy, such that (see [GZ] for details):

h0 = F , hλ(N−) ∩ N = ∅, hλ(N) ∩ N+ = ∅ h1 = (A, 0), where A is a matrix s.t. Bu ⊂ ABu

JISD2012 Geometric methods for manifolds I. 28 May 2012 15 / 27

Page 49: Geometric methods for invariant manifolds in dynamical ... · Geometric methods for invariant manifolds in dynamical systems I. Fixed points and periodic orbits Maciej Capiński AGH

Topological coveringF : Rn → Rn

N = Bu ×Bs

N− = ∂Bu ×BsN+ = Bu × ∂Bs

Definition (covering)

N F⇒ NπuF (N−) ∩Bu = ∅πsF (N) ⊂ Bs∃q0 ∈ N s.t. F (q0) ∈ intN (∗)

s

u

N

N F⇒ N

TheoremThere exists a point p ∈ N suchthat

F (p) = p

Proof. chalk.(*) stronger conditions needed: ∃h : [0, 1]× N → Rn , homotopy, such that (see [GZ] for details):

h0 = F , hλ(N−) ∩ N = ∅, hλ(N) ∩ N+ = ∅ h1 = (A, 0), where A is a matrix s.t. Bu ⊂ ABu

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Topological coveringF : Rn → Rn

N = Bu ×Bs

N− = ∂Bu ×BsN+ = Bu × ∂Bs

Definition (covering)

N F⇒ NπuF (N−) ∩Bu = ∅πsF (N) ⊂ Bs∃q0 ∈ N s.t. F (q0) ∈ intN (∗)

s

u

N

N F⇒ N

TheoremThere exists a point p ∈ N suchthat

F (p) = p

Proof. chalk.(*) stronger conditions needed: ∃h : [0, 1]× N → Rn , homotopy, such that (see [GZ] for details):

h0 = F , hλ(N−) ∩ N = ∅, hλ(N) ∩ N+ = ∅ h1 = (A, 0), where A is a matrix s.t. Bu ⊂ ABu

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Correctly aligned windows

N1

N0

N2N3

N4

Theorem

N0F⇒ N1

F⇒ . . .F⇒ N0

Then there exists a periodic orbitpassing through the sets.

Proof.

N = N0 × . . .×NkF(x0, . . . , xk) =

(F (xk),F (x0), . . . ,F (xk−1))

N F⇒ N

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Correctly aligned windows

N1

N0

N2N3

N4

Theorem

N0F⇒ N1

F⇒ . . .F⇒ N0

Then there exists a periodic orbitpassing through the sets.

Proof.

N = N0 × . . .×NkF(x0, . . . , xk) =

(F (xk),F (x0), . . . ,F (xk−1))

N F⇒ N

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ChaosTwo correctly aligned windows

N0F⇒ N1

N0F⇒ N0

N1F⇒ N0

N1F⇒ N1

N0 N1

For any sequence of zeros and ones we have

0 0 1 0 1 1 . . .

F 0(p) ∈ N0 F 1(p) ∈ N0 F 2(p) ∈ N1 F 3(p) ∈ N0 F 4(p) ∈ N1 F 5(p) ∈ N1 . . .

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ChaosHorseshoe

N0F⇒ N1

N0F⇒ N0

N1F⇒ N0

N1F⇒ N1

N0 N1

Orbits of any prescribed sequences of zeros and ones.

Periodic orbits of any period. For example:

N0F⇒ N1

F⇒ N1F⇒ N1

F⇒ N0

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Example of applicationThe three body problem

x

y

Two planets rotate on circular orbits around center of massThird, small, massless particle does not influence their motion

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N0F⇒ N0 N1

F⇒ N0

N0F⇒ N1 N1

F⇒ N1

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The three body problemFixed points

x

y

We can position a satellite in 5 points and it will remain motionless

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N0F⇒ N0 N1

F⇒ N0

N0F⇒ N1 N1

F⇒ N1

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The three body problemThe problem is three dimensional

x

y

Momenta coordinate

Coordinates x , y , x , y

(Conservation of energy reduces the dimension by one)

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N0F⇒ N0 N1

F⇒ N0

N0F⇒ N1 N1

F⇒ N1

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The three body problemPeriodic orbits

x

y

Momenta coordinate

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N0F⇒ N0 N1

F⇒ N0

N0F⇒ N1 N1

F⇒ N1

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The three body problemWhere is the map and windows?

x

y

Momenta coordinate

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N0F⇒ N0 N1

F⇒ N0

N0F⇒ N1 N1

F⇒ N1

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The three body problemTransition from one window to another

x

y

Momenta coordinate

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N0F⇒ N0 N1

F⇒ N0

N0F⇒ N1 N1

F⇒ N1

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The three body problemChaos in celestial mechanics

x

y

Momenta coordinate

For any sequence of zeros and ones we have

0 0 1 0 1 1 . . .

F 0(p) ∈ N0 F 1(p) ∈ N0 F 2(p) ∈ N1 F 3(p) ∈ N0 F 4(p) ∈ N1 F 5(p) ∈ N1 . . .

[WZ] D. Wilczak, P. Zgliczyński, Comm. Math. Phys. 2003, 2005

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N0F⇒ N0 N1

f⇒ N0

N0F⇒ N1 N1

f⇒ N1

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Closing remarks

We can:

compute fixed points

compute periodic orbits

prove chaotic dynamics

next lectures

Thank you for your attention

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References

Interval Newton method:[N] A. Neumeier, Interval methods for systems of equations. Cambridge University Press, 1990.

Covering relations:[GZ] M. Gidea, P.Zgliczyński, Covering relations for multidimensional dynamical systems I, J. of Diff. Equations,

202(2004) 32–58

3 body problem example:[WZ] D. Wilczak, P.Zgliczyński, Heteroclinic Connections between Periodic Orbits in Planar Restricted Circular Three

Body Problem - A Computer Assisted Proof, Comm. Math. Phys. 234 (2003) 1, 37-75.

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