numerical tools for the computation of homoclinic/heteroclinic orbits

7/30/2019 Numerical tools for the computation of homoclinic/heteroclinic orbits

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Numerical tools

Numerical tools for the computation of

homoclinic/heteroclinic orbits

E. Barrabs

1

J.M. Mondelo

2

M. Oll

3

1Dept. Informtica i Matemtica Aplicada

Universitat de Girona

2Dept. de Matemtiques

Universitat Autnoma de Barcelona

3Dept. de Matemtica Aplicada IUniversitat Politcnica de Catalunya

New trends in astrodynamics and applications, Milan, June 30, July 1, 2,

2008
http://find/http://goback/


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Numerical tools

Outline

Context: the RTBP and the collinear points, motivation and applications Computation of objects: periodic orbits (PO), their invariant manifolds

Homoclinic connections of PO: detection and continuation of families

Some results
http://find/


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Numerical tools

Libration Point orbits

The spatial, circular RTBP

General framework. The RTBP

r2

r1

3m = 0

m =2

m = 11

P =( 1, 0, 0)2

L2

L1

L3

P =(x,y,z)

O

1

P =( , 0, 0)

y

z

x

Differential equations of the

Spatial Circular RTBP:

x = px + y, px = H/x,

y = py x, py = H/y,z = pz, pz = H/z,

Mass ratio: =m2

m1 + m2.

Hamiltonian:

H(x,y,z,pz,py,pz) = 12

(p2x + p2y + p2z ) xpy + ypx 1 r1

r2

with r1 =

(x )2 + y2 + z2, r2 =

(x + 1)2 + y2 + z2.Jacobi constant:

C = 2H+ (1 )
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Numerical tools



Collinear Libration Points

The linear behavior near the collinear equilibrium points is of the type

H2 = x1y1 +p2

(x22 + y22) +

v2

(x23 + y23),

Linear: saddle center center

Non-linear: planar p.o. vertical p.o.

Cantor set of 2D tori

The saddle component makes direct numerical simulations problematic.

y01

y02

y03

y04
http://find/


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Numerical tools



Libration point orbits

Planar and vertical Lyapunov orbits

-0.855-0.85

-0.845-0.84

-0.835-0.83

-0.825

-0.06-0.04

-0.020

0.020.04

0.06

-0.06

-0.04

-0.02

0

0.02

0.040.06

z

xy

z
http://goforward/http://find/http://goback/


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Numerical tools




From vertical to planar: Lissajous orbits

-0.855-0.85

-0.845-0.84

-0.835-0.83

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-0.045-0.03

-0.0150

0.0150.03

0.045

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0.02

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z

xy

z
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Numerical tools





-0.855-0.85

-0.845-0.84

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0.045

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-0.855-0.85

-0.845-0.84

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0.045

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Numerical tools





-0.855-0.85

-0.845-0.84

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0.045

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N i l l
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-0.855-0.85

-0.845-0.84

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-0.825

-0.045-0.03

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0.0150.03

0.045

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Numerical tools





-0.855-0.85

-0.845-0.84

-0.835-0.83

-0.825

-0.045-0.03

-0.0150

0.0150.03

0.045

-0.06

-0.04

-0.02

0

0.02

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xy

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Numerical tools
http://find/


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Numerical tools



Invariant manifolds associated with LPO

-1.05-1

-0.95-0.9

-0.85-0.8-0.08-0.06

-0.04-0.02

00.02

0.040.060.08

-0.02-0.015-0.01

-0.0050

0.0050.01

0.0150.020.025

z

x

y

z


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Motivation: Applications ofL1 and L2

Mission analysis:

Construct new spacecraft trajectories (low cost).

The orbit of comet Oterma in the Sun-Jupiter rotating system followsclosely the invariant manifolds ofL1 and L2.

Using Conley-McGehee tubes, homoclinic orbits allow to prescribe

itineraries between the interior and exterior regions of a moon-planet

system.

Petit Grand Tour for the moons of Jupiter. Genesis discovery mission.

Numerical tools
http://find/


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Interest in L3

L3 is responsible for horseshoe motion

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

It is observed in:

Janus and Epimetheus, coorbitals of Saturn, (Llibre & O, A & A 2001).

Some near Earth Asteriods: 2002 AA29

Numerical tools
http://find/http://find/


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Ambitious goal: Description (as global as possible) of a neighborhood (as

large as possible) of the collinear points including homoclinic and

heteroclinic phenomena.

Particular goal: computation and continuation of homoclinic connections of

LPO around Li, i = 1, 2, 3 for the planar RTBP.

Semi-analytical procedures.

Numerical methodology

Numerical tools
http://find/


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Homoclinic and heteroclinic connections of LPO

Semi-analytical procedures: normal forms and Lindstedt Poincar

expansions.

Advantage: high order approximation of the invariant objects.

Drawback: only for L1 and L2. Developments are useful for small

values of the energy.

Numerical tools
http://find/


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Some papers:

Comet transition (Belbruno & Marsden, Astron J. 1997, Lo and Ross,

JPL IOM 1997)

Existence of heteroclinic connections between orbits around L1 and L2in the planar RTBP (Koon et al, Chaos J. 2000)

Study of homo and heteroclinic connections between planar Lyapunov

p.o. in the planar RTBP (Canalias & Masdemont, DCDS 2006)

Several individual homo and heteroclinic connections between lissajous

and/or quasihalo orbits in the spatial RTBP (Gmez et al.

Nonlinearity, 2004).

Numerical tools
http://find/


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Numerical approach

Direct computation of

Poincar sections:

periodic orbits (step 1) invariant manifolds (step 2)

Wu/s j detection of homoclinic orbits (step 3)

Continuation of families of homoclinic orbits (step 4)

Tools valid for any Li, i = 1, 2, 3 and able to reach higher values of theenergy

Numerical tools
http://find/


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Computation of objects

Periodic orbits

1. Periodic orbits. The system of equations

Let {g(x) = 0} define the Poincar

section

H be the Hamiltonian of the RTBP

T its timeT flow.

p.o.

x Poincare sect.

Unknowns:

h, T,x,y,z,px,py,pz =:x

Equations:

H(x) = hg(x) = 0T(x) = x

Fix h: the solution is a point (i.c. of the p.o. in the Poincar section).

Leth

free: the solution is the curve of i.c. on the fam. of p.o.

Numerical tools
http://find/


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Periodic orbits

Multiple shooting

High instability can be coped with using multiple shooting: search for

several i.c. in the p.o. in order to reduce integration time.

Unknowns:

h, T, x0, x1, . . . , xm1

Equations:

H(x0) = hg(x0) = 0

T/m(x0) = x1T

/m(x1) = x2

...

T/m(xm1) = x0

Numerical tools

C i f bj
http://find/


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Periodic orbits

2., 3., 4. HomoclinicsGeneral procedure:

X invariant hyperbolic object, Wu/s(X) invariant manifolds (twobranches)

fixed section

Wu/s(X) k

homoclinic orbit Wu i Wu j

X

Wu

Ws

Numerical tools

Comp tation of objects
http://find/


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Homoclinic connections ofLi , i = 1, 2, 3

Example. Homoclinic orbits connecting collinear points

invariant manifolds are 1 dimensional

exploration varying

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

= 0.0010015432 = 0.012143988024852

= {x = 1/2}, L3

Numerical tools

http://find/


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Homoclinic orbits to Lyapunov orbits

2. Invariant manifolds of planar Lyapunov orbits


exploration for a fixed value of and energy h

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

-1 -0.5 0 0.5 1

h = 1.50041149280247 h = 1.49952788314423

= {x = 1/2},L3, = SJ

Numerical tools

http://find/


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2. Invariant manifolds of planar Lyapunov orbits


exploration for a fixed value of and energy h

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

-1.2 -1.15 -1.1 -1.05 -1 -0.95 -0.9-6

-4

-2

0

2

4

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4

= {x = 1} = {x = 0}h = 1.58463932117689 h = 1.57485608553295

L2 and = EM

Numerical tools

http://find/


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3. Finding homoclinic orbits. Section curves Wu/s k

X planar Lyapunov orbit, Wu/s(X) invariant manifolds

S = Wu/s(X) k is 1 dimensional

H = h is constant S represented in a plane

-0.467

-0.466

-0.465

-0.464

-0.463

-0.462

-0.461

-0.46

-0.459

-0.929 -0.928 -0.927 -0.926 -0.925 -0.924 -0.923

suitable h

= {x = 1/2}, (y,py)-plane, h = 1.500474767L3 and = SJ

Numerical tools

http://find/


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p j


Intersections Wu/s k

-0.8

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-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2

h = 1.52753129112358 h = 1.50625846902364

Curves Wu 1 and Ws 2

= {x = 0},(px,py) planeL2 and = EM

Numerical tools



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Numerical computation of a section curve

Given (, ) a parametrisation of the inv. manifold

To compute the intersection between the manifold and : {g(x

) = 0}, weneed to continue the solution curve of

g((, )) = 0

in (, ).

Numerical tools

http://find/


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Families of homoclinic orbits: continuation method

4. Families of homoclinic orbits: continuation method

Fixed a p.o. given by an initial condition x0, compute a homoclinic

orbit using the variables

u, s, Tu, Ts

x0

s

Ts

Tu

u

Numerical tools

http://find/


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Continuation of a family of homoclinic orbits

Idea: put all the unknowns into the equations

the energy

the periodic orbit and its period

the eigenvalues and eigenvectors

u

,

s

Tu, Ts

Numerical tools

http://find/


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To find or to continuate an homoclinic connection of a p.o., we can solve

H(x) h = 0g1(x) = 0

T(x) x = 0

vu2 1 = 0 vs2 1 = 0

DT(x)vu uvu = 0 DT(x)vs svs = 0

g2

Tu

u(u, small)

= 0

g2

Ts

s(s, small)

= 0

Tu

u(u, small)

Ts

s(s, small)

= 0

+ multiple shooting if necessary

h as a parameter: we find a family of homoclinic connections

Numerical tools

Results
http://find/


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Homoclinic connections of L.O. around L3

Connections of Lyapunov orbits around L3 ( = SJ)

-1.1

-1.05

-1

-0.95

-0.9

-1.502 -1.499 -1.496 -1.493 -1.49

yf

h

Hn1 Hn2

Examples

Numerical tools

Results
http://find/


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Homoclinic connections to L.O. around L1 and L2

Connections to Lyapunov orbits around L2 ( = EM)

loops around Moon

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-1.59 -1.58 -1.57 -1.56 -1.55 -1.54 -1.53 -1.52

Hi11

Hi21

Hi12

Hi32

Hi42

Examples

loops around Moon and Earth

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

-1.56 -1.55 -1.54 -1.53 -1.52 -1.51 -1.5 -1.49 -1.48

Ho11

Ho21

Ho31

Ho41

Ho51

Ho61

Ho71

Ho81

Ho91

Ho101

Ho111

Ho121

Examples

Numerical tools

Results

S d W k i
http://find/http://find/http://find/


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Summary and Work in progress

Summary

Numerical tools that allow to describe the dynamics related to Li

parameterizations of invariant manifolds

homoclinic (and heteroclinic) connections

some results obtained

Work in progres

heteroclinic connections between LPO

homoclinic/heteroclinic connections of invariant tori

Numerical tools

Results

References
http://find/


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References

E. Belbruno & B. Marsden.

Resonance hopping in cometsAstron. J., 1997.

M. Lo & S. Ross.

Surfing in the Solar System: invariant manifolds and the dynamics of

the Solar System

JPL IOM 312/97, 2-4, 1997.

M. Lo & S. Ross.

Low energy interplanetary transfers using invariant manifolds of L1, L2and halo orbits

AAS-AIAA Space flight meeting, Monterey, 1998.

W.S. Koon et al.

Heteroclinic connections between periodic orbits and resonance

transitions in celestial mechanics

Chaos J., 2000.

Numerical tools

Results

References
http://find/


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References

A. Jorba & J.M. Masdemont.

Dynamics in the center manifold of the collinear points of the restricted

three body problem

Phys. D., 1999.

G. Gmez & J.M. Mondelo.

The dynamics around the collinear equilibrium points of the RTBP

Phys. D., 2001.

G. Gmez el al.

Connecting orbits and invariant manifolds in the spatial restricted

three-body problem

Nonlinearity, 2004.G. Gmez, M. Marcote, & J.M. Mondelo.

The invariant manifold structure of the spatial Hills problem

Dyn. Syst., 2005.

Numerical tools

Results

References
http://find/


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References

E. Barrabs & M.Oll.

Invariant manifolds of L3 and horseshoe motion in the restrictedthree-body problem

Nonlinearity, 2006

E. Canalies & J.M. Masdemont.

Homoclinic and heteroclinic transfer trajectories between planar

Lyapunov orbits in the sun-earth and earth-moon systemsDisc. Cont. Dyn. Syst., 2006.

M. Gidea & J.M. Masdemont.

Geometry of homoclinic connections in a planar circular restricted

three-body problem

Int. J. Bif. Chaos, 2006.

E. Barrabs, J.M.M. Mondelo & M. Oll.

Numerical continuation of families of homoclinic connections of

periodic orbits

Preprint, 2008

Numerical tools
http://find/


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-1.2

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-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

-1.5 -1 -0.5 0 0.5 1 1.5

Numerical tools
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-0.2

-0.1

0

0.1

0.2

-1.25 -1.2 -1.15 -1.1 -1.05 -1 -0.95 -0.9

-0.2

-0.1

0

0.1

0.2

-1.25 -1.2 -1.15 -1.1 -1.05 -1 -0.95 -0.9

-0.2

-0.1

0

0.1

0.2

-1.25 -1.2 -1.15 -1.1 -1.05 -1 -0.95 -0.9

-0.2

-0.1

0

0.1

0.2

-1.25 -1.2 -1.15 -1.1 -1.05 -1 -0.95 -0.9

Numerical tools


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-1.5

-1

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0.5

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1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3
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numerical tools for the computation of homoclinic/heteroclinic orbits

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