Overview of CDS 140B: Introduction to Dynamics · Heteroclinic Connections between Periodic Orbits and Reso-nance Transitions in Celestial Mechanics, Chaos, 10, 427-469. ... • periodic

Overview of CDS 140B:

Introduction to Dynamics

Wang Sang KoonControl and Dynamical Systems, Caltech

[email protected]

1

� Textbook, Other Book and Papers

I Textbook

• Stephen Wiggins [2003]:Introduction to Applied Nonlinear Dynamical Systems andChaos, Second Edition.

I Other Books and Paper

• Ferdinand Verhulst:Nonlinear Differential Equations and Dynamical Systems.

• Lawrence Perko:Differential Equations and Dynamical Systems.

• Koon, W.S., M.W. Lo, J.E. Marsden and S.D. Ross [2000]:Heteroclinic Connections between Periodic Orbits and Reso-nance Transitions in Celestial Mechanics, Chaos, 10, 427-469.

� Goals and Description of CDS 140B

I A continuation of CDS 140A.80% covers basic tools from nonlinear dynamics

• perturbation theory and method of averaging;

• bifurcation theory;

• global bifurcation and chaos;

• Hamiltonian systems.

I Besides standard examples(van der Pol, Duffing, and Lorenz equations),20% applies dynamical system tools in space mission design

• periodic and quasi-periodic orbits,

• invariant manifolds,

• homoclinic and heteroclinic connections,

• symbolic dynamics and chaos.

I Examples: Genesis Discovery Mission andLow Energy Tour of Multiple Moons of Jupiter

� Outline of Presentation

I Main Theme

• how to use dynamical systems theory of 3-body problemin space mission design.

I Background and Motivation:

• NASA’s Genesis Discovery Mission.

• Jupiter Comets.

I Planar Circular Restricted 3-Body Problem.

I Major Results on Tube Dynamics.

I Lobe Dynamics & Navigating in Phase Space.

• A Low Energy Tour of Jupiter’s Moons.

I Conclusion and Ongoing Work.

• Two Full Body Problem and Astroid Pairs.

• Looking into Chemical Reaction Dynamics.

� Acknowledgement

I H. Poincare, J. Moser

I C. Conley, R. McGehee, D. Appleyard

I C. Simo, J. Llibre, R. Martinez

I G. Gomez, J. Masdemont

I B. Farquhar, D. Dunham

I E. Belbruno, B. Marsden, J. Miller

I K. Howell, B. Barden, R. Wilson

I S. Wiggins, L. Wiesenfeld, C. Jaffe, T. Uzer

I V. Rom-Kedar, F. Lekien, L. Vela-Arevalo

I M. Dellnitz, O. Junge, and Paderborn group

� Genesis Discovery Mission

I Genesis spacecraft

• is collecting solar wind samples from a L1 halo orbit,

• will return them to Earth later this year for analysis.

I Halo orbit, transfer/ return trajectories in rotating frame.

L1 L2

x (km)

y(k

m)

-1E+06 0 1E+06

-1E+06

-500000

0

500000

1E+06




I Follows natural dynamics, little propulsion after launch.

I Return-to-Earth portion utilizes heteroclinic dynamics.

L1 L2

x (km)

y(k

m)

-1E+06 0 1E+06

-1E+06

-500000

0

500000

1E+06

� Jupiter Comets

� Jupiter Comets

I Rapid transition from outside to inside Jupiter’s orbit.

I Captured temporarily by Jupiter during transition.

I Exterior (2:3 resonance). Interior (3:2 resonance).

-8 -6 -4 -2 0 2 4 6 8

-8

-6

-4

-2

0

2

4

6

8

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Jupiter’s orbit

Sun

END

START

First encounter

Second encounter

3:2 resonance

2:3 resonance

Oterma’s orbit

Jupiter

� Jupiter Comets

I Belbruno and B. Marsden [1997]

I Lo and Ross [1997]

• Comet in rotating frame follows invariant manifolds.

I Jupiter comets make resonance transition near L1 and L2.

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

x (AU, Sun-Jupiter rotating frame)

y (A

U, S

un-J

upite

r rot

atin

g fr

ame)

L4

L5

L2L1L3J

Jupiter’s orbit-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

-6

-4

-2

0

2

4

6

8

10

S J


y (A

U, S

un-J

upite

r rot

atin

g fr

ame)

Oterma’s orbit

END

START

S

� Planar Circular Restricted 3-Body Problem

I PCR3BP is a good starting model:

• Comets mostly heliocentric, buttheir perturbation dominated by Jupiter’s gravitation.

• Their motion nearly in Jupiter’s orbital plane.

• Jupiter’s small eccentricity plays little role during transition.

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

x (nondimensional units, rotating frame)

y (n

ondi

men

sion

al u

nits

, rot

atin

g fr

ame)

mS = 1 - µ mJ = µ

S J

Jupiter's orbit

L2

L4

L5

L3 L1

comet

� Planar Circular Restricted 3-Body Problem

I 2 main bodies: Sun and Jupiter.

• Total mass normalized to 1: mJ = µ, mS = 1− µ.

• Rotate about center of mass, angular velocity normalized to 1.

I Choose a rotating coordinate system with (0, 0) at center of mass,S and J fixed at (−µ, 0) and (1− µ, 0).

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1


y (n

ondi

men

sion

al u

nits

, rot

atin

g fr

ame)

mS = 1 - µ mJ = µ

S J

Jupiter's orbit

L2

L4

L5

L3 L1

comet

� Equilibrium Points (PCR3BP)

I Comet’s equations of motion are

x− 2y = −∂U

∂xy + 2x = −∂U

∂yU = −x2 + y2

2− 1− µ

rs− µ

rj

I Five equilibrium points:

• 3 unstable equilbrium points on S-J line, L1, L2, L3.

• 2 equilateral equilibrium points, L4, L5.

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1


y (n

ondi

men

sion

al u

nits

, rot

atin

g fr

ame)

mS = 1 - µ mJ = µ

S J

Jupiter's orbit

L2

L4

L5

L3 L1

comet

� Hill’s Realm (PCR3BP)

I Energy integral: E(x, y, x, y) = (x2 + y2)/2 + U(x, y).

I E can be used to determine (Hill’s ) realm in position spacewhere comet is energetically permitted to move.

I Effective potential: U(x, y) = −x2+y2

2 − 1−µrs− µ

rj.

� Hill’s Realm (PCR3BP)

I To fix energy value E is to fix height of plot of U(x, y).Contour plots give 5 cases of Hill’s realm.

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1

-1 0 1

-1

0

1

S J S J

S JS J

Case 1 : C>C1 Case 2 : C1>C>C2

Case 4 : C3>C>C4=C5Case 3 : C2>C>C3

� The Flow near L1 and L2

I For energy value just above that of L2,Hill’s realm contains a “neck” about L1 & L2.

I Comet can make transition through these equilibrium realms.

I 4 types of orbits:periodic, asymptotic, transit & nontransit.

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1


y (n

ondi

men

sion

al u

nits

, rot

atin

g fr

ame)

S JL1

Zero velocity curve bounding forbidden

region


y (n

ondi

men

sion

al u

nits

, rot

atin

g fr

ame)

L2

(a) (b)

exteriorregion

interiorregion

Jupiterregion

forbiddenregion

L2

� The Flow near L1 and L2: Linearization

I [Moser] All the qualitative results of the linearized equations carryover to the full nonlinear equations.

I Recall equations of PCR3BP:

x = vx, vx = 2vy + Ux,

y = vy, vy = −2vx + Uy. (1)

I After linearization,

x = vx, vx = 2vy + ax,

y = vy, vy = −2vx − by. (2)

I Eigenvalues have the form ±λ and ±iν.

I Corresponding eigenvectors are

u1 = (1,−σ, λ,−λσ),

u2 = (1, σ,−λ,−λσ),

w1 = (1,−iτ, iν, ντ ),

w2 = (1, iτ,−iν, ντ ).

� The Flow near L1 and L2: Linearization

I After linearization & making eigenvectors as new coordinateaxes, equations assume simple form

ξ = λξ, η = −λη, ζ1 = νζ2, ζ2 = −νζ1,

with energy function El = ληξ + ν2(ζ2

1 + ζ22).

I The flow near L1, L2 have the form of saddle×center.

η−ξ=

−c

η−ξ=

+c

η−ξ=

0η+ξ=0

|ζ|2 =0

ξ η

|ζ|2 =ρ∗

|ζ| 2=ρ ∗|ζ|2 =0

� Appearance of Orbits in Position Space

I 4 types of orbits: Shown are

• A periodic orbit.

• A typical aymptotic orbit.

• 2 transit orbits.

• 2 non-transit orbits.

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1


y (n

ondi

men

sion

al u

nits

, rot

atin

g fr

ame)

S JL1

Zero velocity curve bounding forbidden

region


y (n

ondi

men

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al u

nits

, rot

atin

g fr

ame)

L2

(a) (b)

exteriorregion

interiorregion

Jupiterregion

forbiddenregion

L2

� Invariant Manifolds as Separatrix

I Stable and unstable manifold tubes act asseparatrices for the flow in R:

• Those inside the tubes are transit orbits.

• Those outside of the tubes are non-transit orbits.

0.88 0.9 0.92 0.94 0.96 0.98 1

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

x (rotating frame)

y (r

otat

ing

fram

e)

L1 PeriodicOrbit

StableManifold

UnstableManifold

Jupiter

StableManifold Forbidden Region

Forbidden RegionUnstableManifold

Moon

� Computation of Periodic Orbit & Invariant Manifolds

I Periodic orbit can be computed by Lindstedt-Poincare method.

I Invariant manifold can be computed by findingeigenvectors of monodromy matrix.

� Major Result (A): Heteroclinic Connection

I Found heteroclinic connection between pair of periodic orbits.

I Found a large class of orbits near this (homo/heteroclinic) chain.

I 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

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� Major Result (A): Heteroclinic Connection

� Review of Horseshoe Dynamics: Pendulum

I Homoclinic orbits separate librational motion fromrotational motion.

� Review of Horseshoe Dynamics: Forced Pendulum

I Periodic forcing leads to transversal intersection andchaotic dynamics.

I For any bi-infinite sequence of itenarary (..., R, L; R,R, ...),there is an orbit whose whereabout matches the sequence.

� Major Result (B): Existence of Transitional Orbits

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

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

I Can even specify number of revolutions the comet makesaround Sun & Jupiter (plus L1 & L2).

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� Major Result (C): Numerical Construction of Orbits

I Developed procedure to construct orbitwith prescribed itinerary.

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

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� Details: Construction of (J,X;J,S,J) Orbits

I Invariant manifold tubes separate transit from nontransit orbits.

I Green curve (Poincare cut of L1 stable manifold).Red curve (cut of L2 unstable manifold).

I Any point inside the intersection region ∆J is a (X;J,S) orbit.

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-0.08

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x (rotating frame)

y (r

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JL1 L2

0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

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y (rotating frame)

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� Details: Construction of (J,X;J,S,J) Orbits

I The desired orbit can be constructed by

• Choosing appropriate Poincare sections and

• linking invariant manifold tubes in right order.

X

S J

U3

U2

U1U4

� Petit Grand Tour of Jupiter’s Moons

I Used invariant manifoldsto construct trajectories with interesting characteristics:

• Petit Grand Tour of Jupiter’s moons.1 orbit around Ganymede. 4 orbits around Europa.

• A ∆V nudges the SC fromJupiter-Ganymede system to Jupiter-Europa system.

I Instead of flybys, can orbit several moons for any duration.

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Ganymede

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Jupiter

Jupiter

Europa’sorbit

Ganymede’sorbit

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Europa

Jupiter




I Jupiter-Ganymede-Europa-SC 4-body system approximatedas 2 coupled 3-body systems

I Invariant manifold tubes of spatial 3-body systemsare linked in right order to construct orbit with desired itinerary.

I Initial solution refined in 4-body model.

Jupiter Europa

Ganymede

Spacecrafttransfertrajectory

∆V at transferpatch point

Jupiter Europa

Ganymede

� Intersection of Tubes: Poincare Section

I Poincare section: Vary configuration of the moons until,

• (x, x)-plane: Intersection found!

• (x, y)-plane: Velocity discontinuity since energies are different.

• A rocket burn ∆V of this magnitude will make transfer trajec-tory “jump” from one tube to the other.

-1.5 -1.45 -1.4 -1.35 -1.3 -1.25 -1.2 -1.15 -1.1 -1.05

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

x (nondimensional, Jupiter-Europa rotating frame)

x (n

ondi

men

sion

al, J

upite

r-E

urop

a ro

tatin

g fr

ame)

.

Ganymede’s L1 periodic orbit unstable manifold

Europa’s L2 periodic orbit

stable manifold

Intersectionpoint

1.55 1.5 1.45 1.4 1.35 1.3 1.25 1.2 1.15 1.1 1.05

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x (nondimensional, Jupiter-Europa rotating frame)

y (n

ondi

men

sion

al, J

upite

r-E

urop

a ro

tatin

g fr

ame)

.

Ganymede’s L1 periodic orbit unstable manifold

Europa’s L2 periodic orbit

stable manifold

Intersectionpoint (and ∆V)

(a) (b)

� Look for Natural Pathways to Bridge the Gap

I Tubes of two 3-body systems may not intersect for awhile.May need large ∆V to “jump” from one tube to another.

I Look for natural pathways to bridge the gap

• between z0 where tube of one system entersand z2 where tube of another system exits (into Europa realm)by “hopping” through phase space.

L1 L2

Exterior Realm

Interior (Jupiter) Realm

Moon Realm

Forbidden Realm(at a particular energy level)

spacecraft

Poincare sectionL1 L2

f1

f2

f12

f2

f1z0

z1z2

z3z4

z5 U2

U1

Exit

Entrance

(a) (b)

� Transport in Phase Space via Tube & Lobe Dynamics

I By using

• tubes of rapid transition that connect realms

• lobe dynamics to hop through phase space,

New tour only needs ∆V = 20m/s (50 times less).

f1

f2

f12

f2

f1z0

z1z2

z3z4

z5 U2

U1

Exit

Entrance

Low Energy Tour of Jupiter’s MoonsSeen in Jovicentric Inertial Frame

Jupiter

Callisto Ganymede Europa

(a) (b)

� Tube Dynamics: Mixed Phase Space

I Poincare section reveals mixed phase space:

• resonance regions and

• ‘‘chaotic sea”.

L1 L2

Exterior Realm

Interior (Jupiter) Realm

Moon Realm

Forbidden Realm(at a particular energy level)

spacecraft

Poincare sectionL1 L2

Identify

Argument of Periapse (radians)Semim

ajo

r Axis

(Neptune =

1)

(a) (b)

� Transport between Regions via Lobe Dynamics

I Invariant manifolds divide phase space into resonance regions.

I Transport between regions can be studied via lobe dynamics.

Identify

Argument of Periapse

Semim

ajo

r Axis

R1

R2

q0

pipj

f -1(q0)q1

L2,1(1)

L1,2(1)

f (L1,2(1))

f (L2,1(1))

� Transport between Regions via Lobe Dynamics

I Segments of unstable and stable manifoldsform partial barriers between regions R1 and R2.

I L1,2(1), L2,1(1) are lobes; they form a turnstile.

• In one iteration, only points from R1 to R2 are in L1,2

• only points from R2 to R1 are in L2,1(1).

I By studying pre-images of L1,2(1),one can find efficient way from R1 to R2.

R1

R2

q0

pipj

f -1(q0)q1

L2,1(1)

L1,2(1)

f (L1,2(1))

f (L2,1(1))

� Hopping through Resonaces in Low Energy Tour

I Guided by lobe dynamics, hopping through resonances(essential for low energy tour) can be performed.

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0.7

0.75

0.8

P

m1

m2

Surface-of-section Large orbit changes

� Tube/Lobe Dynamics: Transport in Solar System

I To use tube dynamics/lobe dynamics of spatial 3-body problemto systematically design low-fuel trajectory.

I Part of our program to study transport in solar systemusing tube and lobe dynamics.

f1

f2

f12

f2

f1z0

z1z2

z3z4

z5 U2

U1

Exit

Entrance

Low Energy Tour of Jupiter’s MoonsSeen in Jovicentric Inertial Frame

Jupiter

Callisto Ganymede Europa

(a) (b)

� Tube/Lobe Dynamics: 2FBP/Asteroid Pairs

I To study dynamical interacton between 2 rigid bodieswhere their rotational and translational motions are coupled.

• formation of binary asteroids (shown below are Ida and Dactyl)

• evolution of asteroid rotational states

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5


y (

nondim

ensi

onal

unit

s, r

ota

ting f

ram

e)

Exterior Realm

InteriorRealm

U2

Poincare Section

InteriorRealm

U1

Poincare Section

Asteroid

Energetically

Forbidden Realm

(a) (b)

� Multi-scale Dynamics of Biomolecules (ICB)

I Carry out model reduction (POD) for simple biomolecules,keeping chains of molecules and more complex systems in view.

I Indentify conformations in simple molecular modelsusing techniques of AIS and set oriented methods.

I Compute transport rates between different conformationsby both lobe dynamics and set oriented methods.

I Collaborate with Institute of Collaborative Bio-Technologycd (ICB).

Documents

Overview of CDS 140B: Introduction to Dynamics · Heteroclinic Connections between Periodic Orbits and Reso-nance Transitions in Celestial Mechanics, Chaos, 10, 427-469. ... • periodic