Dynamical Systems

and Space Mission DesignJerrold Marsden, Wang Koon and Martin Lo

Wang Sang KoonControl and Dynamical Systems, Caltech

koon@cds.caltech.edu

� Global Orbit Structure: Outline

� Outline of Lecture 3B:

• Construction of Poincare map.

• Review of Horseshoe Dynamics.

• Symbolic Dynamics for the PCR3BP.

• Main Theorem on Global Orbit Structure.

� Global Orbit Structure: Overview

� Found heteroclinic connection between pair of periodic orbits.

� Find a large class of orbits near this (homo/heteroclinic) chain.

� Comet can follow these channels in rapid transition.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Sun

Jupiter’s Orbit

x (AU, Sun-Jupiter rotating frame)

y (A

U, S

un-J

upite

r ro

tatin

g fr

ame)

x (AU, Sun-Jupiter rotating frame)

y (A

U, S

un-J

upite

r ro

tatin

g fr

ame)

Jupiter

SunHomoclinicOrbit

Homoclinic Orbit

L1 L2L1 L2

HeteroclinicConnection

� Global Orbit Structure: Overview

� Symbolic sequence used to label itinerary of each comet orbit.

� Main Theorem: For any admissible itinerary,e.g., (. . . ,X,J,S,J,X, . . . ), there exists an orbit whosewhereabouts matches this itinerary.

� Can even specify number of revolutions the comet makesaround L1 & L2 as well as Sun & Jupiter.

S J

L1 L2

X

� Global Orbit Structure: Overview

� Using the proof of Main Theorem as the guide, we developprocedure to construct orbit with prescribed itinerary.

� Example: An orbit with itinerary (X,J;S,J,X).

� Petit Grand Tour of Jovian moons & Shoot the Moon.

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

x (rotating frame)

y (r

otat

ing

fram

e)

0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Jupiter

Forbidden Region

Forbidden Region

L1 L2

Sun

x (rotating frame)

y (r

otat

ing

fram

e)

(X,J,S,J,X)Orbit

� Global Orbit Structure: Energy Manifold

� Schematic view of energy manifold.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Sun

Jupiter’s Orbit

x (AU, Sun-Jupiter rotating frame)

y (A

U, S

un-J

upite

r ro

tatin

g fr

ame)

x (AU, Sun-Jupiter rotating frame)

y (A

U, S

un-J

upite

r ro

tatin

g fr

ame)

Jupiter

SunHomoclinicOrbit

Homoclinic Orbit

L1 L2L1 L2

HeteroclinicConnection

A1

B1

D1

C1

A1' B1'

E1

F1

H1

G1

A2' B2'A2

B2

D2

C2

G1' H1'

E2

F2

H2

G2

G2' H2'

n1,1 n1,2 n2,1 n2,2

a1,1+

a1,1− a1,2

+

a1,2− a2,1

+

a2,1− a2,2

+

a2,2−

U3

U2

U4U1

interiorregion (S )

exteriorregion (X )

Jupiterregion (J )

L1 equilibriumregion (R1)

L2 equilibriumregion (R2)

p1,1

p1,2

p2,1

p2,2y = 0

x = 1−µ

x = 1−µ

y = 0

� Global Orbit Structure: Poincare Map

� Reducing study of global orbit structure to study of discrete map.

A1

B1

D1

C1

A1' B1'

E1

F1

H1

G1

A2' B2'A2

B2

D2

C2

G1' H1'

E2

F2

H2

G2

G2' H2'

n1,1 n1,2 n2,1 n2,2

a1,1+

a1,1− a1,2

+

a1,2− a2,1

+

a2,1− a2,2

+

a2,2−

U3

U2

U4U1

interiorregion (S )

exteriorregion (X )

Jupiterregion (J )

L1 equilibriumregion (R1)

L2 equilibriumregion (R2)

p1,1

p1,2

p2,1

p2,2y = 0

x = 1−µ

x = 1−µ

y = 0

U3

U2

U4U1interiorregion

exteriorregion

Jupiterregion

A1'

B1'

C1

D1

U1

E1

F1

A2'

B2'

U3

C2

D2

G1'

H1'

U2

E2

F2

G2'

H2'

U4

y = 0

x = 1−µ

x = 1−µ

y = 0

� Construction of Poincare Map

� Construct Poincare map P (tranversal to the flow)whose domain U consists of 4 squares Ui.

� Squares U1 and U4 contained in y = 0,each centers around a transversal homoclinic point.

� Squares U2 and U3 contained in x = 1 − µ,each centers around a transversal heteroclinic point.

A1

B1

D1

C1

A1' B1'

E1

F1

H1

G1

A2' B2'A2

B2

D2

C2

G1' H1'

E2

F2

H2

G2

G2' H2'

n1,1 n1,2 n2,1 n2,2

a1,1+

a1,1− a1,2

+

a1,2− a2,1

+

a2,1− a2,2

+

a2,2−

U3

U2

U4U1

interiorregion (S )

exteriorregion (X )

Jupiterregion (J )

L1 equilibriumregion (R1)

L2 equilibriumregion (R2)

p1,1

p1,2

p2,1

p2,2y = 0

x = 1−µ

x = 1−µ

y = 0

� Global Orbit Structure near the Chain

� Consider invariant set Λ of points in U whose images and pre-images under all iterations of P remain in U .

Λ = ∩∞n=−∞Pn(U).

� Invariant set Λ contains all recurrent orbits near the chain.It provides insight into the global dynamics around the chain.

� Chaos theory told us to first consider only the first forward andbackward iterations:

Λ1 = P−1(U) ∩ U ∩ P 1(U).

U3

U2

U4U1interiorregion

exteriorregion

Jupiterregion

A1'

B1'

C1

D1

U1

E1

F1

A2'

B2'

U3

C2

D2

G1'

H1'

U2

E2

F2

G2'

H2'

U4

y = 0

x = 1−µ

x = 1−µ

y = 0

� Review of Horseshoe Dynamics: Pendulum

� Review of Horseshoe Dynamics: Forced Pendulum

� Review of Horseshoe Dynamics: First Iteration

� Review of Horseshoe Dynamics: First Iteration

� Review of Horseshoe Dynamics: Second Iteration

� Review of Horseshoe Dynamics: Second Iteration

...0,0; ...1,0; ...1,1; ...0,1;

;1,0...

;1,1...

;0,1...

;0,0...

...1,0;1,1...

D

� Conley-Moser Conditions: Horseshoe-type Map

� For horseshoe-type map h satisfying Conley-Moser conditions,the invariant set of all iterations, Λh = ∩∞

n=−∞hn(Q),can be constructed and visualized in a standard way.

• Strip condition: h maps “horizontal strips” H0, H1to “vertical strips” V0, V1, (with horizontal boundaries to hor-izontal boundaries and vertical boundaries to vertical bound-aries).

• Hyperbolicity condition: h has uniform contractionin horizontal direction and expansion in vertical direction.

...0,0; ...1,0; ...1,1; ...0,1;

;1,0...

;1,1...

;0,1...

;0,0...

...1,0;1,1...

D

� Conley-Moser Conditions: Horseshoe-type Map

� Invariant set of first iterations Λ1h = h−1(D) ∩ D ∩ h1(D)

has 4 squares, with addresses (0; 0), (1; 0), (1; 1), (0; 1).

� Invariant set of second iterations has 16 squarescontained in 4 squares of first stage.

� This process can be repeated ad infinitumdue to Conley-Moser Conditions.

� What remains is invariant set of points Λh which are in 1-to-1corr. with set of bi-infinite sequences of 2 symbols (. . . , 0; 1, . . . ).

...0,0; ...1,0; ...1,1; ...0,1;

;1,0...

;1,1...

;0,1...

;0,0...

...1,0;1,1...

D

� Horizontal (H31n ) & Vertical (V 21

n ) Strips

� Recall: image of abutting arc of stable manifold cutspiral infinitely many times towards unstable manifold cut.

ΓL1,1s,S

ΓL1,1u,S

A1'

B1'

C1

D1

U1

E1

F1

A2'

B2'

U3

C2

G1'

H1'

U2

p1,1

ΓL1,1u,J

D2

ΓL2,1s,J

ΓL2,1u,J

p1,2

ΓL1,1s,J

VH1,0'

HE1,0

VH1,n'

e5

e6

e8

e7

p2,1

e9

HE1,n

e10 e11

e12

P −1

P

e2

e1

e3

e4

D1 A1'

U

C1 A1'

U

D1 B1'U

n1

r1−

b1

a1−

δ3

δ2

Q2

ψ2(γ2)d1

−

ψ3(γ3)

Q3

r1+

d1+

a1+

γ1

P1

γ3

P3

r2+

n2

b2

a2+

γ4

P4

γ2

P2

d2+

r2−

d2−

a2−

δ1

Q1

ψ1(γ1)

δ4

Q4

ψ4(γ4)

� Horizontal (Hijn ) & Vertical (V

jim ) Strips

� Hence, U ∩ P−1(U) has 8 families of horizontal strips Hijn .

• Each point in horizontal strip Hijn is in Ui and will wind n

times around an equilibrium point before reaching Uj.

• We can associate each point in Hijn both an address in U

and an itinerary (; ui, n, uj).

A1' B1'

C1

D1

H2’

Hn11

Hn12

Vm13 Vm

11

C2

D2

C1

D1

A2' B2' A1' B1'

H1' G1'

C2

D2

Hn23

Hn24

Vm21 Vm

23

H2’

A2' B2'

E1

F1

Hn31

Hn32

Vm34 Vm

32

H2’

G2' H2'

E2

F2

Hn43

Hn44

Vm44 Vm

42

U1

U2

U3

U4Qm;n

3;12

� Horizontal (Hijn ) & Vertical (V

jim ) Strips

� Similarly, U ∩ P 1(U) consists of 8 families of vertical strips Vjim .

• Each point in vertical strip Vjim came from Ui and has wound

m times around an equilibrium point before arriving at Uj.

• We can associate each point in Vjim both an address in U

and an itenerary (ui, m; uj).

A1' B1'

C1

D1

H2’

Hn11

Hn12

Vm13 Vm

11

C2

D2

C1

D1

A2' B2' A1' B1'

H1' G1'

C2

D2

Hn23

Hn24

Vm21 Vm

23

H2’

A2' B2'

E1

F1

Hn31

Hn32

Vm34 Vm

32

H2’

G2' H2'

E2

F2

Hn43

Hn44

Vm44 Vm

42

U1

U2

U3

U4Qm;n

3;12

� Invariant Set Λ1 under First Interations

� The set Λ1 = P−1(U) ∩ U ∩ P 1(U)is intersections of all horizontal and vertical strips.

• Each point of Q3;12m;n = H12

n ∩ V 13m has an itenerary

(u3, m; u1, n, u2) which is a concatenation of (u1; n, u2) (H12n )

and (u3, m; u1) (V 13m ).

A1' B1'

C1

D1

H2’

Hn11

Hn12

Vm13 Vm

11

C2

D2

C1

D1

A2' B2' A1' B1'

H1' G1'

C2

D2

Hn23

Hn24

Vm21 Vm

23

H2’

A2' B2'

E1

F1

Hn31

Hn32

Vm34 Vm

32

H2’

G2' H2'

E2

F2

Hn43

Hn44

Vm44 Vm

42

U1

U2

U3

U4Qm;n

3;12

� Application of Symbolic Dynamics

� Labeling “squares” Q3;12m;n with (u3, m; u1, n, u2) is in line with

characterizing orbits via bi-infinite sequences of “symbols”.

� To keep track of itenerary w.r.t. 4 squares Ui, we use subshiftwith 4 symbols ui, (. . . , u3; u1, u2, . . . ), and a transition matrix

A =

1 1 0 00 0 1 11 1 0 00 0 1 1

.

A1' B1'

C1

D1

U1

U2

H1' G1'

C2

D2

U3

A2' B2'

E1

F1

U4

G2' H2'

E2

F2

� Application of Symbolic Dynamics

� To keep track of number of revolutions around L1 or L2,we use full shift with integer symbols, (. . . , m; n, . . . ).

� ‘‘Squares” Qi;jkm;n in 1-to-1 corr. with sequences (ui, m; uj, n, uk).

� Symbolic sequence is used to label address of each “square”and identifies itenerary of its orbits.

� To generalize beyond first iteration, need to review chaos theory.

A1' B1'

C1

D1

H2’

Hn11

Hn12

Vm13 Vm

11

C2

D2

C1

D1

A2' B2' A1' B1'

H1' G1'

C2

D2

Hn23

Hn24

Vm21 Vm

23

H2’

A2' B2'

E1

F1

Hn31

Hn32

Vm34 Vm

32

H2’

G2' H2'

E2

F2

Hn43

Hn44

Vm44 Vm

42

U1

U2

U3

U4Qm;n

3;12

� Conley-Moser Conditions: Horseshoe-type Map

� For horseshoe-type map h satisfying Conley-Moser conditions,the invariant set of all iterations, Λh = ∩∞

n=−∞hn(Q),can be constructed and visualized in a standard way.

• Strip condition: h maps “horizontal strips” H0, H1to “vertical strips” V0, V1, (with horizontal boundaries to hor-izontal boundaries and vertical boundaries to vertical bound-aries).

• Hyperbolicity condition: h has uniform contractionin horizontal direction and expansion in vertical direction.

...0,0; ...1,0; ...1,1; ...0,1;

;1,0...

;1,1...

;0,1...

;0,0...

...1,0;1,1...

D

� Conley-Moser Conditions: Horseshoe-type Map

� Invariant set of first iterations Λ1h = h−1(D) ∩ D ∩ h1(D)

has 4 squares, with addresses (0; 0), (1; 0), (1; 1), (0; 1).

� Invariant set of second iterations has 16 squarescontained in 4 squares of first stage.

� This process can be repeated ad infinitumdue to Conley-Moser Conditions.

� What remains is invariant set of points Λh which are in 1-to-1corr. with set of bi-infinite sequences of 2 symbols (. . . , 0; 1, . . . ).

...0,0; ...1,0; ...1,1; ...0,1;

;1,0...

;1,1...

;0,1...

;0,0...

...1,0;1,1...

D

� Generalized Conley-Moser Conditions

� Proved P satisfies Generalized Conley-Moser conditions:

• Strip condition: it maps “horizontal strips” Hijn

to “vertical strips” Vjin .

• Hyperbolicity condition: it has uniform contractionin horizontal direction and expansion in vertical direction.

A1' B1'

C1

D1

U1

U2

H1' G1'

C2

D2

U3

A2' B2'

E1

F1

U4

G2' H2'

E2

F2

� Generalized Conley-Moser Conditions

� Shown are invariant set Λ1 under first iteration.

� Since P satisfies Generalized Conley-Moser Conditions,this process can be repeated ad infinitum.

� What remains is invariant set of points Λ which are in 1-to-1corr. with set of bi-infinite sequences (. . . , ui, m; uj, n, uk, . . . ).

A1' B1'

C1

D1

H2’

Hn11

Hn12

Vm13 Vm

11

C2

D2

C1

D1

A2' B2' A1' B1'

H1' G1'

C2

D2

Hn23

Hn24

Vm21 Vm

23

H2’

A2' B2'

E1

F1

Hn31

Hn32

Vm34 Vm

32

H2’

G2' H2'

E2

F2

Hn43

Hn44

Vm44 Vm

42

U1

U2

U3

U4Qm;n

3;12

� Global Orbit Structure: Main Theorem

� Main Theorem: For any admissible itinerary,e.g., (. . . ,X,1,J,0;S,1,J,2,X, . . . ), there exists an orbit whosewhereabouts matches this itinerary.

� Can even specify number of revolutions the comet makesaround Sun & Jupiter.

S J

L1 L2

X

� Global Orbit Structure: Dynamical Channels

� Found a large class of orbits near homo/heteroclinic chain.

� Comet can follow these channels in rapid transition.

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Sun

Jupiter’s Orbit

x (AU, Sun-Jupiter rotating frame)

y (A

U, S

un-J

upite

r ro

tatin

g fr

ame)

x (AU, Sun-Jupiter rotating frame)

y (A

U, S

un-J

upite

r ro

tatin

g fr

ame)

Jupiter

SunHomoclinicOrbit

Homoclinic Orbit

L1 L2L1 L2

HeteroclinicConnection

� Construction of Orbits with Prescribed Itinerary

� Using the proof of Main Theorem as the guide, we developprocedure to construct orbit with prescribed itinerary.

� Example: An orbit with itinerary (X,J;S,J,X).

� Petit Grand Tour of Jovian moons & Shoot the Moon.

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

x (rotating frame)

y (r

otat

ing

fram

e)

0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Jupiter

Forbidden Region

Forbidden Region

L1 L2

Sun

x (rotating frame)

y (r

otat

ing

fram

e)

(X,J,S,J,X)Orbit

� Construction of (J,X;J,S,J) Orbits

JS L2

L1

PoincareSection

JL1 L2

(J,X;J,S,J)

(J,X;J)

(;J,S,J)

Forbidden Region

Forbidden Region

Sun

Jupiter

(;X,J)

(J;X)

(;S,J)

(J;S)

∆X = (J;X,J) (J,X;J,S,J)∆S = (J;S,J)

(J,X;J,S,J)Orbit

UnstableManifold

PoincareSection

Exterior Region Interior Region Jupiter Region

∆X = (J;X,J)

∆S = (J;S,J)