We explain a methodology to compute families of homo/heteroclinic connections between periodic orbits. We show the relation of some homoclinic connections with resonant transitions in the RTBP

1. Introduction CRTBP Homo/heteroclinics Resonant transitions
Homoclinic/heteroclinic connections between
periodic orbits and resonant transitions in the RTBP.
E. Barrab´es (UdG) J.M. Mondelo (UAB) M. Oll´e (UPC)
Third Colloquium on Dynamical Systems, Control and Applications.
UAM June 21-23, 2013.
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 1 / 31

2. Introduction CRTBP Homo/heteroclinics Resonant transitions
Outline
Introduction
The Planar Circular Restricted Three Body Problem
Families of homo/heteroclinic orbits
Resonant transitions in the CRTBP
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 2 / 31

3. Introduction CRTBP Homo/heteroclinics Resonant transitions
Motivations and aims
Homoclinic and heteroclinic connections of hyperbolic objects play an im-
portant role in the study of dynamical systems from a global point of view.
To have a better understanding of their structure allow us to:
detect transit/non-transit orbits and trajectories with prescribed
itineraries
design of space missions using the dynamics around equilibrium points
design of low-energy transfers
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 3 / 31

4. Introduction CRTBP Homo/heteroclinics Resonant transitions
Libration point missions
Artemis mission
Artemis P1-spacecraft follows a heteroclinic connection between orbits around
the two Lagrangian points L1 and L2 of the Earth–Moon system
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 4 / 31

5. Introduction CRTBP Homo/heteroclinics Resonant transitions
Trajectories with prescribed itineraries and resonant transitions
Comet Oterma: resonant transitions
(from Koon et al, Chaos (2000))
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 5 / 31

6. Introduction CRTBP Homo/heteroclinics Resonant transitions
Motivations and aims
Aims:
to construct maps of homoclinic and heteroclinic connections in different
scenarios
to develop a numerical methodology that overcomes the convergence
restrictions of semianalytical techniques and automatizes the process
to relate homoclinic–heteroclinic chains with resonance transitions like
the orbits of the Jupiter comet Oterma
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 6 / 31

7. Introduction CRTBP Homo/heteroclinics Resonant transitions
Planar Circular Restricted Three Body Problem (CRTBP)
L1
L2
L5
L4
L3
SE 0.5
0.5
−0.5
−0.5
Primaries:
masses 1 − µ, µ
circular orbits
synodical frame
−→rS = (µ, 0), −→rJ = (µ − 1, 0)
Equilibrium points:
L1, L2, L3 Lagrangian points
L4, L5 triangular points
Energy: h =
−1
2
x2
+ y2
+ 2
1 − µ
r1
+ 2
µ
r2
+ µ(1 − µ) − v2
r2
1 = (x − µ)2
+ y2
, r2
2 = (x − µ + 1)2
+ y2
, v2
= ˙x2
+ ˙y2
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 7 / 31

8. Introduction CRTBP Homo/heteroclinics Resonant transitions
Hill’s region
Zero velocity curves: −2h = x2
+ y2
+ 2
1 − µ
r1
+ 2
µ
r2
+ µ(1 − µ)
-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
y
x
SJ
Consider values of the energy such that exists the zero velocity curve and the
Hill’s region has one connected component.
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 8 / 31

9. Introduction CRTBP Homo/heteroclinics Resonant transitions
Behavior of the Lagrangian points
Linearized equations: ˙z = DF(Lj)z.
center×center×saddle
Spec DF(Lj) = {±iω1, ±iω2, ±λ}
Lyapunov center theorem: two families of periodic orbits (p.o.) are born
at each equilibrium point (Lyapunov orbits)
-0.855 -0.85 -0.845 -0.84 -0.835 -0.83 -0.825 -0.05-0.04-0.03-0.02-0.01 0 0.010.020.030.040.05
-0.05
-0.025
0
0.025
0.05
The lyapunov orbits inherit the hyperbolic behavior: there exists
invariant manifolds Wu/s
associated to them
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 9 / 31

10. Introduction CRTBP Homo/heteroclinics Resonant transitions
Invariant manifolds of Lyapunov orbits
-2.5
-1.5
-0.5
0.5
1.5
2.5
-1.5 -0.5 0.5 1.5 2.5
y
x
SJ
-0.1
-0.05
0
0.05
0.1
-1.1 -1.05 -1 -0.95 -0.9
y x
Outer and inner regions Jupiter’s region
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 10 / 31

11. Introduction CRTBP Homo/heteroclinics Resonant transitions
Computation of families of homo/heteroclinic orbits
Methodology
X, Y invariant hyperbolic objects, Wu/s
(X), Wu/s
(Y ) invariant
manifolds
Σ fixed Poincar´e section
homoclinic orbit −→ Wu
(X) ∩ Σj
∩ Ws
(X) ∩ Σk
heteroclinic orbit −→ Wu
(X) ∩ Σj
∩ Ws
(Y ) ∩ Σk
Σ θs
θu
x0
Σ
θu
xu
0
θs
xs
0
homoclinic to X heteroclinic from X to Y
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 11 / 31

12. Introduction CRTBP Homo/heteroclinics Resonant transitions
Numerical procedure for periodic orbits
Steps for the computation of homo/heteroclinic orbits to periodic orbits (p.o.)
1 computation of the p.o. and their invariant manifolds, and detection of
existence of homo/heteroclinic orbits
-0.04
-0.02
0
0.02
0.04
-1.08 -1.04 -1 -0.96 -0.92
y
x
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.05 -0.04 -0.03 -0.02 -0.01 0
py
y
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 12 / 31

13. Introduction CRTBP Homo/heteroclinics Resonant transitions
Numerical procedure for periodic orbits
Steps for the computation of homo/heteroclinic orbits to periodic orbits (p.o.)
2 computation of a single homo/heteroclinic orbit
a system of equations whose solution is the homo/heteroclinic solution is
solved
3 continuation of families of homo/heteroclinic orbits
the system of equations is numerically continued by a standard
predictor-corrector method
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 13 / 31

14. Introduction CRTBP Homo/heteroclinics Resonant transitions
System of equations



H(x1) − h = 0, H(x2) − h = 0,
p(x1) = 0, p(x2) = 0,
φT1
(x1) − x1 = 0, φT2
(x2) − x2 = 0,
vu 2
− 1 = 0, vs 2
− 1 = 0,
DφT1
(x1)vu
− Λu
vu
= 0, DφT2
(x2)vs
− Λs
vs
= 0,



g φT u ψu
1 (θu
, ξ) = 0,
g φ−T s ψs
2(θs
, ξ) = 0,
φT u ψu
1 (θu
, ξ) − φ−T s ψs
2(θs
, ξ) = 0,
2(2n + 3) + n + 2 equations, 4n + 6 unknowns multiple shooting system
solved by a minimum`ınorm, least`ısquares Newton correction procedure.
The instability of the orbits is coped with a multiple shooting method
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 14 / 31

15. Introduction CRTBP Homo/heteroclinics Resonant transitions
Heteroclinic connections between two p.o. around L1 and L2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
-1.1 -1.05 -1 -0.95 -0.9
y
x
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
-1.1 -1.05 -1 -0.95 -0.9
y x
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 15 / 31

16. Introduction CRTBP Homo/heteroclinics Resonant transitions
Homoclinic connections between a p.o. around L1
Homoclinics in the inner region
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
y
x
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
y
x
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
y
x
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 16 / 31

17. Introduction CRTBP Homo/heteroclinics Resonant transitions
Homoclinic connections between a p.o. around L2 (outer)
Homoclinics in the outer region
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1.5 -1 -0.5 0 0.5 1 1.5 2
y
x
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1.5 -1 -0.5 0 0.5 1 1.5 2
y
x
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1.5 -1 -0.5 0 0.5 1 1.5 2
y
x
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 17 / 31

18. Introduction CRTBP Homo/heteroclinics Resonant transitions
Families of homoclinic connections
-0.95
-0.9
-0.85
-0.8
-0.75
-0.7
-0.65
-0.6
-0.55
-1.52 -1.515 -1.51 -1.505 -1.5 -1.495
y
h
Hi1
Hi3 Hi2
Hi4
Hi5
Hi6
Hi7
Hi8
Hi9
Hi10
Hi11
Hi12
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
-1.52 -1.515 -1.51 -1.505 -1.5y
h
Ho1÷4
Ho5÷8
Ho9÷12
inner outer
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 18 / 31

19. Introduction CRTBP Homo/heteroclinics Resonant transitions
Introduction
The Planar Circular Restricted Three Body Problem
Families of homo/heteroclinic orbits
Resonant transitions in the CRTBP
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 19 / 31

20. Introduction CRTBP Homo/heteroclinics Resonant transitions
Outer, inner and Jupiter’s regions
We consider the Sun-Jupiter CRTBP for energy levels such that the outer and
inner regions are connected
-2.5
-1.5
-0.5
0.5
1.5
2.5
-1.5 -0.5 0.5 1.5 2.5
y
x
SJ
-0.1
-0.05
0
0.05
0.1
-1.1 -1.05 -1 -0.95 -0.9
y
x
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 20 / 31

21. Introduction CRTBP Homo/heteroclinics Resonant transitions
Dynamical chains / channels
Koon, Lo, Marsden, and Ross. Heteroclinic connections between periodic
orbits and resonance transitions in celestial mechanics. (Chaos, 2000)
Dynamical chain: sequence of homo, hetero, homo visiting different
regions (itinerary)
Dynamical channel: set of orbits following the same itinerary. The
dynamical chain is its backbone.
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 21 / 31

22. Introduction CRTBP Homo/heteroclinics Resonant transitions
Transit and non-transit orbits
A trajectory approaching a p.o., either forward or backward in time from one
of the three regions, is considered transit if it traverses the bottleneck
corresponding to the LPO and goes to the next region
x axis
W u
− W
s
+
W u
+W s
−
x axis
W u
− W s
+
W u
+W
s
−
Transit orbit Non-transit orbit
Transit orbits are known to lie in the interior of the invariant manifold tubes
of the p.o., that separate them from non–transit orbits (Conley,1968; McGehee
1969)
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 22 / 31

23. Introduction CRTBP Homo/heteroclinics Resonant transitions
Transit through the Jupiter’s region
-0.1
-0.05
0
0.05
0.1
-1.1 -1.05 -1 -0.95 -0.9
y
x
Fast transit: transit through the Jupiter’s region. The trajectory must lie in
the interior of the adequate branches of all the invariant manifolds associated
to two p.o. around L2 and L1.
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 23 / 31

24. Introduction CRTBP Homo/heteroclinics Resonant transitions
Transit from the inner region
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
y
x
S
J
-0.4
-0.2
0
0.2
0.4
0.8 1 1.2 1.4 1.6 1.8
py
px
Transit from Jupiter region → inner region → Jupiter region:
the orbits may lie in the interior of both invariant manifolds Wu
and Ws
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 24 / 31

25. Introduction CRTBP Homo/heteroclinics Resonant transitions
Transit: inner orbits
-0.4
-0.2
0
0.2
0.4
0.8 1 1.2 1.4 1.6 1.8
py
px
-0.17
-0.15
-0.13
1.21 1.23 1.25 1.27
-0.4
-0.2
0
0.2
0.4
0.8 1 1.2 1.4 1.6 1.8
py
px
-0.4
-0.2
0
0.2
0.4
0.8 1 1.2 1.4 1.6 1.8
py
px
−→ h increasing −→
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 25 / 31

26. Introduction CRTBP Homo/heteroclinics Resonant transitions
Transit: outer orbits
-0.25
-0.15
-0.05
0.05
0.15
0.25
-1.1 -0.9 -0.7 -0.5
py
px
0.15
0.155
0.16
0.165
-0.65 -0.64 -0.63
-0.25
-0.15
-0.05
0.05
0.15
0.25
-1.1 -0.9 -0.7 -0.5
py
px
-0.25
-0.15
-0.05
0.05
0.15
0.25
-1.1 -0.9 -0.7 -0.5
py
px
−→ h increasing −→
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 26 / 31

27. Introduction CRTBP Homo/heteroclinics Resonant transitions
Resonances
Resonances are defined in terms of two body dynamics:
An elliptic (keplerian) orbit is p : q resonant with Jupiter, if it performs p
revolutions around the Sun while Jupiter performs q revolutions.
The mean motion equals a−3/2
= p/q, being a the semimajor axis that
can be calculated as
a−1
=
2
r
− v2
For trajectories of the CRTBP that behave essentially as a two-body
solution a will be approximately constant.
the orbits on Wu
, Ws
for the homoclinics that provide the dynamical chains
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 27 / 31

28. Introduction CRTBP Homo/heteroclinics Resonant transitions
Transit orbits and resonances
-0.4
-0.2
0
0.2
0.4
0.8 1 1.2 1.4 1.6 1.8
py
px
-0.162
-0.157
-0.152
-0.147
-0.142
1.235 1.24 1.245 1.25 1.255
py
px
1
2
3
4
Wu 1
W
s 2
1.494
1.498
1.502
1.506
1.51
1.235 1.24 1.245 1.25 1.255
a-3/2
px
1
2
34
Transit orbits around a 3:2 resonance
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 28 / 31

29. Introduction CRTBP Homo/heteroclinics Resonant transitions
Families of homoclinic connections: resonances
Families of inner orbits
-0.95
-0.9
-0.85
-0.8
-0.75
-0.7
-0.65
-0.6
-0.55
-1.52 -1.515 -1.51 -1.505 -1.5 -1.495
y
h
Hi1
Hi3 Hi2
Hi4
Hi5
Hi6
Hi7
Hi8
Hi9
Hi10
Hi11
Hi12
1.2
1.25
1.3
1.35
1.4
1.45
1.5
1.55
-1.52 -1.515 -1.51 -1.505 -1.5 -1.495
a-3/2 h
Hi1
Hi3,4
Hi2
Hi5
Hi6
Hi7,8
Hi9
Hi10
Hi11,12
Resonances at 3:2, 4:3, 5:4
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 29 / 31

30. Introduction CRTBP Homo/heteroclinics Resonant transitions
Families of homoclinic connections: resonances
Families of outer orbits
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
-1.52 -1.515 -1.51 -1.505 -1.5
y
h
Ho1÷4
Ho5÷8
Ho9÷12
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
-1.52 -1.515 -1.51 -1.505 -1.5
a-3/2
h
Ho1÷4
Ho5÷8
Ho9÷12
Resonances at 3:4, 2:3, 1:2
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 30 / 31

31. Introduction CRTBP Homo/heteroclinics Resonant transitions
Conclusions
We present a methodology for the numerical computation of families of
homoclinics and heteroclinic connections to hyperbolic periodic orbits
a higher values of the energy can be reached with respect semi-analytical
procedures
automatization of the continuation
We have explored the relation between such families with resonant
transitions in the CRTBP
We have determined ranges of energy in which they are possible, and
enlarged the choice of resonances that can be connected
Barrab´es,Mondelo,Oll´e () Homo/heteroclinics and resonant transitions DySCA III 31 / 31