homo/heteroclinic connections between periodic orbits

Introduction CRTBP Homo/heteroclinics Resonant transitions

Homoclinic/heteroclinic connections betweenperiodic orbits and resonant transitions in the RTBP.

E. Barrabes (UdG) J.M. Mondelo (UAB) M. Olle (UPC)

Third Colloquium on Dynamical Systems, Control and Applications.UAM June 21-23, 2013.

Barrabes,Mondelo,Olle () Homo/heteroclinics and resonant transitions DySCA III 1 / 31


Outline

Introduction

The Planar Circular Restricted Three Body Problem

Families of homo/heteroclinic orbits

Resonant transitions in the CRTBP



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 prescribeditineraries

design of space missions using the dynamics around equilibrium points

design of low-energy transfers



Libration point missions

Artemis mission

Artemis P1-spacecraft follows a heteroclinic connection between orbits aroundthe two Lagrangian points L1 and L2 of the Earth–Moon system



Trajectories with prescribed itineraries and resonant transitions

Comet Oterma: resonant transitions

(from Koon et al, Chaos (2000))



Motivations and aims

Aims:

to construct maps of homoclinic and heteroclinic connections in differentscenarios

to develop a numerical methodology that overcomes the convergencerestrictions of semianalytical techniques and automatizes the process

to relate homoclinic–heteroclinic chains with resonance transitions likethe orbits of the Jupiter comet Oterma



Planar Circular Restricted Three Body Problem (CRTBP)

L 1L 2

L 5

L 4

L 355 5

5

5

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− µr1

+ 2µ

r2+ µ(1− µ)− v2

)r21 = (x− µ)2 + y2, r22 = (x− µ+ 1)2 + y2, v2 = x2 + y2



Hill’s region

Zero velocity curves: −2h = x2 + y2 + 21− µr1

+ 2µ

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 theHill’s region has one connected component.



Behavior of the Lagrangian points

Linearized equations: z = DF (Lj)z.

center×center×saddle

SpecDF (Lj) = {±iω1,±iω2,±λ}Lyapunov center theorem: two families of periodic orbits (p.o.) are bornat 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 0.01 0.02 0.03 0.04 0.05-0.05

-0.025

0

0.025

0.05

The lyapunov orbits inherit the hyperbolic behavior: there existsinvariant manifolds Wu/s associated to them



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 -0.9y

x

Outer and inner regions Jupiter’s region



Computation of families of homo/heteroclinic orbits

Methodology

X, Y invariant hyperbolic objects, Wu/s(X), Wu/s(Y ) invariantmanifolds

Σ fixed Poincare section

homoclinic orbit −→(Wu(X) ∩ Σj

)∩(W s(X) ∩ Σk

)heteroclinic orbit −→

(Wu(X) ∩ Σj

)∩(W s(Y ) ∩ Σk

)

Σ θs

θu x0

Σ

θu

xu0θs

xs0

homoclinic to X heteroclinic from X to Y



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 ofexistence 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

p y

y



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 issolved

3 continuation of families of homo/heteroclinic orbits

the system of equations is numerically continued by a standardpredictor-corrector method



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 − Λuvu = 0, DφT2

(x2)vs − Λsvs = 0,g(φTu

(ψu1 (θu, ξ)

))= 0,

g(φ−T s

(ψs2(θs, ξ)

))= 0,

φTu

(ψu1 (θu, ξ)

)− φ−T s

(ψs2(θs, ξ)

)= 0,

2(2n+ 3) + n+ 2 equations, 4n+ 6 unknowns multiple shooting systemsolved by a minimumınorm, leastısquares Newton correction procedure.

The instability of the orbits is coped with a multiple shooting method



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 -0.9y

x



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



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



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

Hi6Hi7

Hi8

Hi9Hi10

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



Introduction

The Planar Circular Restricted Three Body Problem

Families of homo/heteroclinic orbits

Resonant transitions in the CRTBP



Outer, inner and Jupiter’s regions

We consider the Sun-Jupiter CRTBP for energy levels such that the outer andinner 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



Dynamical chains / channels

Koon, Lo, Marsden, and Ross. Heteroclinic connections between periodicorbits and resonance transitions in celestial mechanics. (Chaos, 2000)

Dynamical chain: sequence of homo, hetero, homo visiting differentregions (itinerary)

Dynamical channel: set of orbits following the same itinerary. Thedynamical chain is its backbone.



Transit and non-transit orbits

A trajectory approaching a p.o., either forward or backward in time from oneof the three regions, is considered transit if it traverses the bottleneckcorresponding 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 tubesof the p.o., that separate them from non–transit orbits (Conley,1968; McGehee1969)



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 inthe interior of the adequate branches of all the invariant manifolds associated

to two p.o. around L2 and L1.



Transit from the inner region

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

y

x

SJ

-0.4

-0.2

0

0.2

0.4

0.8 1 1.2 1.4 1.6 1.8

p y

px

Transit from Jupiter region → inner region → Jupiter region:the orbits may lie in the interior of both invariant manifolds Wu and W s



Transit: inner orbits

-0.4

-0.2

0

0.2

0.4

0.8 1 1.2 1.4 1.6 1.8

p y

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

p y

px

-0.4

-0.2

0

0.2

0.4

0.8 1 1.2 1.4 1.6 1.8

p y

px

−→ h increasing −→



Transit: outer orbits

-0.25

-0.15

-0.05

0.05

0.15

0.25

-1.1 -0.9 -0.7 -0.5

p y

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

p y

px

-0.25

-0.15

-0.05

0.05

0.15

0.25

-1.1 -0.9 -0.7 -0.5

p y

px

−→ h increasing −→



Resonances

Resonances are defined in terms of two body dynamics:

An elliptic (keplerian) orbit is p : q resonant with Jupiter, if it performs prevolutions around the Sun while Jupiter performs q revolutions.

The mean motion equals a−3/2 = p/q, being a the semimajor axis thatcan be calculated as

a−1 =2

r− v2

For trajectories of the CRTBP that behave essentially as a two-bodysolution a will be approximately constant.

the orbits on Wu, W s

for the homoclinics that provide the dynamical chains



Transit orbits and resonances

-0.4

-0.2

0

0.2

0.4

0.8 1 1.2 1.4 1.6 1.8

p y

px

-0.162

-0.157

-0.152

-0.147

-0.142

1.235 1.24 1.245 1.25 1.255

p y

px

1

2

3

4

Wu 1

Ws 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



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

Hi6Hi7

Hi8

Hi9Hi10

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 -1.495a-3

/2h

Hi1Hi3,4

Hi2

Hi5

Hi6

Hi7,8

Hi9

Hi10

Hi11,12

Resonances at 3:2, 4:3, 5:4



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 -1.5a-3

/2

h

Ho1÷4

Ho5÷8

Ho9÷12

Resonances at 3:4, 2:3, 1:2



Conclusions

We present a methodology for the numerical computation of families ofhomoclinics and heteroclinic connections to hyperbolic periodic orbits

a higher values of the energy can be reached with respect semi-analyticalprocedures

automatization of the continuation

We have explored the relation between such families with resonanttransitions in the CRTBP

We have determined ranges of energy in which they are possible, andenlarged the choice of resonances that can be connected


homo/heteroclinic connections between periodic orbits

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