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7/30/2019 Numerical tools for the computation of homoclinic/heteroclinic orbits
1/40
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
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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
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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
Libration Point orbits
The spatial, circular RTBP
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
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Numerical tools
Libration Point orbits
The spatial, circular RTBP
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
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Numerical tools
Libration Point orbits
The spatial, circular RTBP
Libration point orbits
From vertical to planar: Lissajous orbits
-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
0.04
0.06
z
xy
z
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Numerical tools
Libration Point orbits
The spatial, circular RTBP
Libration point orbits
From vertical to planar: Lissajous orbits
-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
0.04
0.06
z
xy
z
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Numerical tools
Libration Point orbits
The spatial, circular RTBP
Libration point orbits
From vertical to planar: Lissajous orbits
-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
0.04
0.06
z
xy
z
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Numerical tools
Libration Point orbits
The spatial, circular RTBP
Libration point orbits
From vertical to planar: Lissajous orbits
-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
0.04
0.06
z
xy
z
N i l l
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Numerical tools
Libration Point orbits
The spatial, circular RTBP
Libration point orbits
From vertical to planar: Lissajous orbits
-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
0.04
0.06
z
xy
z
N i l t l
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Numerical tools
Libration Point orbits
The spatial, circular RTBP
Libration point orbits
From vertical to planar: Lissajous orbits
-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
0.04
0.06
z
xy
z
Numerical tools
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Numerical tools
Libration Point orbits
The spatial, circular RTBP
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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Numerical tools
7/30/2019 Numerical tools for the computation of homoclinic/heteroclinic orbits
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Numerical tools
Libration Point orbits
The spatial, circular RTBP
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
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Libration Point orbits
The spatial, circular RTBP
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
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Libration Point orbits
The spatial, circular RTBP
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
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Libration Point orbits
The spatial, circular RTBP
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
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Libration Point orbits
The spatial, circular RTBP
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
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Libration Point orbits
The spatial, circular RTBP
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
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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
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Computation of objects
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
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Computation of objects
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
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Computation of objects
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
Computation of objects
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Computation of objects
Homoclinic orbits to Lyapunov orbits
2. Invariant manifolds of planar Lyapunov orbits
invariant manifolds are 2 dimensional
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
Computation of objects
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Computation of objects
Homoclinic orbits to Lyapunov orbits
2. Invariant manifolds of planar Lyapunov orbits
invariant manifolds are 2 dimensional
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
Computation of objects
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Computation of objects
Homoclinic orbits to Lyapunov orbits
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
Computation of objects
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p j
Homoclinic orbits to Lyapunov orbits
Intersections Wu/s k
-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
-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
Computation of objects
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Homoclinic orbits to Lyapunov orbits
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
Computation of objects
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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
Computation of objects
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Families of homoclinic orbits: continuation method
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
Computation of objects
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Families of homoclinic orbits: continuation method
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
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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
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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
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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
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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
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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
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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
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-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
-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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-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
-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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