Continuation of periodic orbits in conservativesystems

Jorge Galán Vioque

Departamento de Matemática Aplicada II &Instituto de Matemáticas de la Universidad de Sevilla (IMUS)

with F. J. Muñoz-Almaraz, E. Freire, E. Doedel and A. Vanderbauwhede

Bifurcation Analysis and its ApplicationsMontreal, July 2010

Jorge Galán Vioque

Unofficial change in the schedule (lunch and discussions)Today at 14:30.....

Germany: n - España: n + 1 Podemos!

Jorge Galán Vioque

Unofficial change in the schedule (lunch and discussions)Today at 14:30.....

Germany: n - España: n + 1 Podemos!

Jorge Galán Vioque

Outline

From the webpage of the conference:

The workshop will include introductory and specializedpresentations.

1 A simple example.2 Implementation in AUTO

3 Continuation in conservative systems or continuationwithout parameters; an alternative to reduction methods.

4 Application to the horseshoe solution of the RTBP(preliminary results).

5 What is missing?

Jorge Galán Vioque

Outline

From the webpage of the conference:

The workshop will include introductory and specializedpresentations.

1 A simple example.

2 Implementation in AUTO

3 Continuation in conservative systems or continuationwithout parameters; an alternative to reduction methods.

4 Application to the horseshoe solution of the RTBP(preliminary results).

5 What is missing?

Jorge Galán Vioque

Outline

From the webpage of the conference:

The workshop will include introductory and specializedpresentations.

1 A simple example.2 Implementation in AUTO

3 Continuation in conservative systems or continuationwithout parameters; an alternative to reduction methods.

4 Application to the horseshoe solution of the RTBP(preliminary results).

5 What is missing?

Jorge Galán Vioque

Outline

From the webpage of the conference:

The workshop will include introductory and specializedpresentations.

1 A simple example.2 Implementation in AUTO

3 Continuation in conservative systems or continuationwithout parameters; an alternative to reduction methods.

4 Application to the horseshoe solution of the RTBP(preliminary results).

5 What is missing?

Jorge Galán Vioque

Outline

From the webpage of the conference:

The workshop will include introductory and specializedpresentations.

1 A simple example.2 Implementation in AUTO

3 Continuation in conservative systems or continuationwithout parameters; an alternative to reduction methods.

4 Application to the horseshoe solution of the RTBP(preliminary results).

5 What is missing?

Jorge Galán Vioque

The best-seller in mathematical modelling

Galileo’s pendulum

3 parameters: L,m,gNewton’s second law:

mLθ + mg sin θ = 0

Jorge Galán Vioque

The best-seller in mathematical modelling

Galileo’s pendulum3 parameters: L,m,g

Newton’s second law:

mLθ + mg sin θ = 0

Jorge Galán Vioque

The best-seller in mathematical modelling

Galileo’s pendulum3 parameters: L,m,gNewton’s second law:

mLθ + mg sin θ = 0

Jorge Galán Vioque

The best-seller in mathematical modeling

Galileo’s pendulum3 parameters: L,m,gNewton’s second law:

θ +gL

sin θ = 0

Jorge Galán Vioque

The best-seller in mathematical modeling

Galileo’s pendulum

Rescaling time with τ =√

Lg .

Newton’s second law:

Galileum Pendulum

θ + sin θ = 0

One dof ODE withoutparameter with two equilibria:θ = 0 (S) and θ = π (U) and aone parameter family ofperiodic orbits.

Jorge Galán Vioque

The best-seller in mathematical modeling

Galileo’s pendulum

Rescaling time with τ =√

Lg .

Newton’s second law:

Galileum Pendulum

θ + sin θ = 0

One dof ODE withoutparameter with two equilibria:θ = 0 (S) and θ = π (U) and aone parameter family ofperiodic orbits.

Jorge Galán Vioque

Phase portrait of Galileo’s pendulum

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4Galileos pendulum

u(1)

u(2)

Jorge Galán Vioque

The reduction method

Position and velocity are not independent of each other.

The system has a first integral or conserved quantity:

E =θ2

2+ 1 − cos θ.

The dimension of the problem can be reduced byeliminating the velocity:

θ =√

2(E − 1 + cosθ).

We have introduced E as an internal parameter that canbe used for continuation.

Jorge Galán Vioque

The reduction method

Position and velocity are not independent of each other.The system has a first integral or conserved quantity:

E =θ2

2+ 1 − cos θ.

The dimension of the problem can be reduced byeliminating the velocity:

θ =√

2(E − 1 + cosθ).

We have introduced E as an internal parameter that canbe used for continuation.

Jorge Galán Vioque

The reduction method

Position and velocity are not independent of each other.The system has a first integral or conserved quantity:

E =θ2

2+ 1 − cos θ.

The dimension of the problem can be reduced byeliminating the velocity:

θ =√

2(E − 1 + cosθ).

We have introduced E as an internal parameter that canbe used for continuation.

Jorge Galán Vioque

The reduction method

Position and velocity are not independent of each other.The system has a first integral or conserved quantity:

E =θ2

2+ 1 − cos θ.

The dimension of the problem can be reduced byeliminating the velocity:

θ =√

2(E − 1 + cosθ).

We have introduced E as an internal parameter that canbe used for continuation.

Jorge Galán Vioque

Geometrical picture: Cylinder Theorem

Jorge Galán Vioque

Geometrical picture: Reduction

Jorge Galán Vioque

Alternative method: undamped pendulum

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4Galileos pendulum

u(1)

u(2)

Jorge Galán Vioque

Alternative method: positive dissipation

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4Galileos pendulum

u(1)

u(2)

Jorge Galán Vioque

Alternative: negative dissipation

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4Galileos pendulum

u(1)

u(2)

Jorge Galán Vioque

The idea: θ − αθ + sin θ = 0

α > 0

α = 0

α < 0

Jorge Galán Vioque

AUTO implementation

y1 = y2y2 = − sin(y1)

y = (y1, y2) = (0,0)

Jorge Galán Vioque

AUTO implementation

y1 = y2y2 = − sin(y1) − αy2

y = (y1, y2) = (0,0)

where α is an "unfolding" parameter.

Task: continue the trivial y = 0 equilibrium in α.Result: α = 0 will be a "vertical" Hopf bifurcation.

Idea: Continue the family of periodic orbits in α with a pseudoarclength scheme.

Jorge Galán Vioque

AUTO implementation

y1 = y2y2 = − sin(y1) − αy2

y = (y1, y2) = (0,0)

where α is an "unfolding" parameter.

Task: continue the trivial y = 0 equilibrium in α.

Result: α = 0 will be a "vertical" Hopf bifurcation.Idea: Continue the family of periodic orbits in α with a pseudo

arclength scheme.

Jorge Galán Vioque

AUTO implementation

y1 = y2y2 = − sin(y1) − αy2

y = (y1, y2) = (0,0)

where α is an "unfolding" parameter.

Task: continue the trivial y = 0 equilibrium in α.Result: α = 0 will be a "vertical" Hopf bifurcation.

Idea: Continue the family of periodic orbits in α with a pseudoarclength scheme.

Jorge Galán Vioque

AUTO implementation

y1 = y2y2 = − sin(y1) − αy2

y = (y1, y2) = (0,0)

where α is an "unfolding" parameter.

Task: continue the trivial y = 0 equilibrium in α.Result: α = 0 will be a "vertical" Hopf bifurcation.

Idea: Continue the family of periodic orbits in α with a pseudoarclength scheme.

Jorge Galán Vioque

AUTO results

−1 −0.5 0 0.5 1−1

0

1

2

3

α

L2no

rm

(a)

−2 0 2−2

−1

0

1

2(d)

x1

x2

0 0.2 0.4 0.6 0.8 1−5

0

5

x1

(e1)

0 0.2 0.4 0.6 0.8 1−2

0

2

x2

scaled time

(e2)

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

step

size

scaled time

(f)

0 0.5 1 1.5 2

102

Per

iod

H

(b)

0 0.5 1 1.5 20

1

2

3

4

L2 n

orm

H

(c)

Jorge Galán Vioque

Remarks

1 It is straightforward to implement

(if we know the unfolding term ) [Physica D 181 (2001)].2 It can be extended to k independent conserved quatities.3 Bifurcations can be detected and followed.4 We can detect homo and heteroclinic connections.5 The computation preserves the simplectic character of the

problem (Hamiltonian case).6 For reversible system there are further simplifications.7 The Galileum pendulum could be added as a demo in the

AUTO documentation if it fulfills the necessary condition.

Auto Demo CharacterizationEvery AUTO demo should have a 3 letter name:

Jorge Galán Vioque

Remarks

1 It is straightforward to implement(if we know the unfolding term ) [Physica D 181 (2001)].

2 It can be extended to k independent conserved quatities.3 Bifurcations can be detected and followed.4 We can detect homo and heteroclinic connections.5 The computation preserves the simplectic character of the

problem (Hamiltonian case).6 For reversible system there are further simplifications.7 The Galileum pendulum could be added as a demo in the

AUTO documentation if it fulfills the necessary condition.

Auto Demo CharacterizationEvery AUTO demo should have a 3 letter name:

Jorge Galán Vioque

Remarks

1 It is straightforward to implement(if we know the unfolding term ) [Physica D 181 (2001)].

2 It can be extended to k independent conserved quatities.

3 Bifurcations can be detected and followed.4 We can detect homo and heteroclinic connections.5 The computation preserves the simplectic character of the

problem (Hamiltonian case).6 For reversible system there are further simplifications.7 The Galileum pendulum could be added as a demo in the

AUTO documentation if it fulfills the necessary condition.

Auto Demo CharacterizationEvery AUTO demo should have a 3 letter name:

Jorge Galán Vioque

Remarks

1 It is straightforward to implement(if we know the unfolding term ) [Physica D 181 (2001)].

2 It can be extended to k independent conserved quatities.3 Bifurcations can be detected and followed.

4 We can detect homo and heteroclinic connections.5 The computation preserves the simplectic character of the

problem (Hamiltonian case).6 For reversible system there are further simplifications.7 The Galileum pendulum could be added as a demo in the

AUTO documentation if it fulfills the necessary condition.

Auto Demo CharacterizationEvery AUTO demo should have a 3 letter name:

Jorge Galán Vioque

Remarks

1 It is straightforward to implement(if we know the unfolding term ) [Physica D 181 (2001)].

2 It can be extended to k independent conserved quatities.3 Bifurcations can be detected and followed.4 We can detect homo and heteroclinic connections.

5 The computation preserves the simplectic character of theproblem (Hamiltonian case).

6 For reversible system there are further simplifications.7 The Galileum pendulum could be added as a demo in the

AUTO documentation if it fulfills the necessary condition.

Auto Demo CharacterizationEvery AUTO demo should have a 3 letter name:

Jorge Galán Vioque

Remarks

1 It is straightforward to implement(if we know the unfolding term ) [Physica D 181 (2001)].

2 It can be extended to k independent conserved quatities.3 Bifurcations can be detected and followed.4 We can detect homo and heteroclinic connections.5 The computation preserves the simplectic character of the

problem (Hamiltonian case).

6 For reversible system there are further simplifications.7 The Galileum pendulum could be added as a demo in the

AUTO documentation if it fulfills the necessary condition.

Auto Demo CharacterizationEvery AUTO demo should have a 3 letter name:

Jorge Galán Vioque

Remarks

1 It is straightforward to implement(if we know the unfolding term ) [Physica D 181 (2001)].

2 It can be extended to k independent conserved quatities.3 Bifurcations can be detected and followed.4 We can detect homo and heteroclinic connections.5 The computation preserves the simplectic character of the

problem (Hamiltonian case).6 For reversible system there are further simplifications.

7 The Galileum pendulum could be added as a demo in theAUTO documentation if it fulfills the necessary condition.

Auto Demo CharacterizationEvery AUTO demo should have a 3 letter name:

Jorge Galán Vioque

Remarks

1 It is straightforward to implement(if we know the unfolding term ) [Physica D 181 (2001)].

2 It can be extended to k independent conserved quatities.3 Bifurcations can be detected and followed.4 We can detect homo and heteroclinic connections.5 The computation preserves the simplectic character of the

problem (Hamiltonian case).6 For reversible system there are further simplifications.7 The Galileum pendulum could be added as a demo in the

AUTO documentation if it fulfills the necessary condition.

Auto Demo CharacterizationEvery AUTO demo should have a 3 letter name:

Jorge Galán Vioque

Remarks

1 It is straightforward to implement(if we know the unfolding term ) [Physica D 181 (2001)].

2 It can be extended to k independent conserved quatities.3 Bifurcations can be detected and followed.4 We can detect homo and heteroclinic connections.5 The computation preserves the simplectic character of the

problem (Hamiltonian case).6 For reversible system there are further simplifications.7 The Galileum pendulum could be added as a demo in the

AUTO documentation if it fulfills the necessary condition.

Auto Demo Characterization

Every AUTO demo should have a 3 letter name:

Jorge Galán Vioque

Remarks

1 It is straightforward to implement(if we know the unfolding term ) [Physica D 181 (2001)].

2 It can be extended to k independent conserved quatities.3 Bifurcations can be detected and followed.4 We can detect homo and heteroclinic connections.5 The computation preserves the simplectic character of the

problem (Hamiltonian case).6 For reversible system there are further simplifications.7 The Galileum pendulum could be added as a demo in the

AUTO documentation if it fulfills the necessary condition.

Auto Demo CharacterizationEvery AUTO demo should have a 3 letter name:

Jorge Galán Vioque

Remarks

1 It is straightforward to implement(if we know the unfolding term ) [Physica D 181 (2001)].

2 It can be extended to k independent conserved quatities.3 Bifurcations can be detected and followed.4 We can detect homo and heteroclinic connections.5 The computation preserves the simplectic character of the

problem (Hamiltonian case).6 For reversible system there are further simplifications.7 The Galileum pendulum could be added as a demo in the

AUTO documentation if it fulfills the necessary condition.

Auto Demo CharacterizationEvery AUTO demo should have a 3 letter name: pen

Name already in use!!!

Jorge Galán Vioque

Remarks

1 It is straightforward to implement(if we know the unfolding term ) [Physica D 181 (2001)].

2 It can be extended to k independent conserved quatities.3 Bifurcations can be detected and followed.4 We can detect homo and heteroclinic connections.5 The computation preserves the simplectic character of the

problem (Hamiltonian case).6 For reversible system there are further simplifications.7 The Galileum pendulum could be added as a demo in the

AUTO documentation if it fulfills the necessary condition.

Auto Demo CharacterizationEvery AUTO demo should have a 3 letter name: pen

Name already in use!!!

Jorge Galán Vioque

Remarks

1 It is straightforward to implement(if we know the unfolding term ) [Physica D 181 (2001)].

2 It can be extended to k independent conserved quatities.3 Bifurcations can be detected and followed.4 We can detect homo and heteroclinic connections.5 The computation preserves the simplectic character of the

problem (Hamiltonian case).6 For reversible system there are further simplifications.7 The Galileum pendulum could be added as a demo in the

AUTO documentation if it fulfills the necessary condition.

Auto Demo CharacterizationEvery AUTO demo should have a 3 letter name: gal

Jorge Galán Vioque

Advanced presentation: continuation of horseshoeorbit in the RTBP

An easy to state but challenging problem with a singleconserved quantity and tons of known families of PO.

x − 2y =∂Ω

∂x,

y − 2x =∂Ω

∂y,

Ω(x , y) =12

(x2 + y2) +(1 − µ)

r1+µ

r2+

12µ(1 − µ),

r21 = (x + µ)2 + y2,

r22 = (x + µ− 1)2 + y2.

Jacobi constant: C = 2Ω(x , y) − x2 − y2.

Jorge Galán Vioque

Advanced presentation: continuation of horseshoeorbit in the RTBP

An easy to state but challenging problem with a singleconserved quantity and tons of known families of PO.

x − 2y =∂Ω

∂x,

y − 2x =∂Ω

∂y,

Ω(x , y) =12

(x2 + y2) +(1 − µ)

r1+µ

r2+

12µ(1 − µ),

r21 = (x + µ)2 + y2,

r22 = (x + µ− 1)2 + y2.

Jacobi constant: C = 2Ω(x , y) − x2 − y2.

Jorge Galán Vioque

RTBP: Lagrange Points

The RTBP has five equilibrium points Li that depend of µ .The Jacobi constant on them (Ci = C(Li)) fulfills therelation

3 = C4 = C5 < C3 < C1 < C2,

and C3 = C1 for µ = 12 .

Jorge Galán Vioque

RTBP: Lagrange Families [IJBC 17 (2007)]

Jorge Galán Vioque

Stability of the Lagrange points: Horseshoe orbit

Jorge Galán Vioque

Horseshoe orbits in the RTBP

A horseshoe periodic orbit is a planar PO in which theparticle surrounds only the equilibrium points L3, L4 and L5and has only two orthogonal crossings with x-axis (reversible).Explain the behavior of two satellites of Saturn: Janus andEpimetheus that rotate in coorbital orbits in a fixed frame.Analyzing the zero velocity curves and for µ ∼ 0 Llibre &Ollé provide a constructive method combining the µ = 0solutions with a shooting method.There are many families of horseshoe families and theyexist for any value of µ.They are numerically difficult to compute (extendedprecision) and continue (stability, near collisions, . . . ).

Jorge Galán Vioque

Horseshoe orbits: µ = 0.000953875 (Taylor 1981)

1.25 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00U(1)

1.5

1.0

0.5

0.0

0.5

1.0

1.5U

(2) 1

Jorge Galán Vioque

Horseshoe orbits: µ = 0.000953875

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1L2-NORM

3.1

3.0

2.9

2.8

2.7

2.6

2.5

2.4

2.3

2.2

2.1

2.0PA

R(3

)

1

2 3

4

5

678

X

Jorge Galán Vioque

Horseshoe orbits: µ = 0.000953875

1.5 1.0 0.5 0.0 0.5 1.0U(1)

1.5

1.0

0.5

0.0

0.5

1.0

1.5U

(2) 2

Jorge Galán Vioque

Horseshoe orbits: µ = 0.000953875

1.5 1.0 0.5 0.0 0.5 1.0 1.5U(1)

1.5

1.0

0.5

0.0

0.5

1.0

1.5U

(2)

3

Jorge Galán Vioque

Horseshoe orbits: µ = 0.000953875

1.5 1.0 0.5 0.0 0.5 1.0 1.5U(1)

1.5

1.0

0.5

0.0

0.5

1.0

1.5U

(2)

4

Jorge Galán Vioque

Horseshoe orbits: µ = 0.000953875

1.5 1.0 0.5 0.0 0.5 1.0 1.5U(1)

1.5

1.0

0.5

0.0

0.5

1.0

1.5U

(2)

5

Jorge Galán Vioque

Horseshoe orbits: µ = 0.000953875

1.5 1.0 0.5 0.0 0.5 1.0 1.5U(1)

1.5

1.0

0.5

0.0

0.5

1.0

1.5U

(2)

8

Jorge Galán Vioque

Horseshoe orbits: µ = 0.000953875

1.5 1.0 0.5 0.0 0.5 1.0 1.5U(1)

1.5

1.0

0.5

0.0

0.5

1.0

1.5U

(2)

10

Jorge Galán Vioque

Horseshoe orbits: µ = 0.000953875

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5U(1)

2.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

2.0U

(2)

17

Jorge Galán Vioque

Horseshoe orbits: µ = 0.000953875

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5U(1)

2.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

2.0U

(2)

20

Jorge Galán Vioque

Horseshoe orbits: µ = 0.000953875

1.2 1.1 1.0 0.9 0.8U(1)

0.2

0.1

0.0

0.1

0.2

0.3U

(2)

Jorge Galán Vioque

Horseshoe orbits: continuation in µ with fixed T

1.5 1.0 0.5 0.0 0.5 1.0 1.5U(1)

1.5

1.0

0.5

0.0

0.5

1.0

1.5U

(2)

12

Jorge Galán Vioque

Horseshoe orbits: continuation in µ with fixed T

1.5 1.0 0.5 0.0 0.5 1.0 1.5U(1)

1.5

1.0

0.5

0.0

0.5

1.0

1.5U

(2)

13

Jorge Galán Vioque

Horseshoe orbits: continuation in µ with fixed T

1.5 1.0 0.5 0.0 0.5 1.0 1.5U(1)

1.5

1.0

0.5

0.0

0.5

1.0

1.5U

(2)

14

Jorge Galán Vioque

Horseshoe orbits: continuation in µ with fixed T

1.5 1.0 0.5 0.0 0.5 1.0 1.5U(1)

1.5

1.0

0.5

0.0

0.5

1.0

1.5U

(2)

15

Jorge Galán Vioque

Horseshoe orbits: continuation in µ with fixed T

1.5 1.0 0.5 0.0 0.5 1.0 1.5U(1)

1.5

1.0

0.5

0.0

0.5

1.0

1.5U

(2)

16

Jorge Galán Vioque

Horseshoe orbits: continuation in µ with fixed T

1.5 1.0 0.5 0.0 0.5 1.0 1.5U(1)

1.5

1.0

0.5

0.0

0.5

1.0

1.5U

(2)

18

Jorge Galán Vioque

Horseshoe orbits: continuation in µ with fixed T

1.5 1.0 0.5 0.0 0.5 1.0 1.5U(1)

1.5

1.0

0.5

0.0

0.5

1.0

1.5U

(2)

19

Jorge Galán Vioque

Horseshoe orbits: nonsymmetrical branch

Jorge Galán Vioque

Stable and unstable manifolds of L3 varying µ

E. Barrabés, J. M. Mondelo and M. Ollé. Celestial Mechanics105 2009.

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP 201

(! j,!k), the mutually symmetric orbit is also an homoclinic connection of type (+ j,+k).We also remark that homoclinic connections of type (! j,+k) or (+ j,!k) are symmetric.

Given µ, in order to check the existence of a homoclinic orbit of a given type (i.e., j, kand signs) we follow this simple method: we take ! = x = c, being c a constant, and weconsider the corresponding points qu

j (µ) = (xu, yu, x "u, y"u) and qsk (µ) = (xs, ys, x "s, y"s)

and the functions:

dy(µ) = yu ! ys, dx "(µ) = x "u ! x "s, dy"(µ) = y"u ! y"s

since xu = xs . Observe that these functions depend on j and k although we have skippedexplicitly this dependence in the notation. We also remark, and this will be seen later on, thatwhen we fix a j and a k, these functions may not be continuous due to the appearance ofloops in the manifolds. Taking into account the direction of the orbits at the intersection of themanifolds with the section and that the energy is the same for both branches, the condition(4) of homoclinic connection is satisfied if two of these functions are equal to zero.

When varying µ some observations with respect to the behavior of the invariant manifoldsare needed.

• As µ increases, the separation between the branches increases and the minimum distanceto the small primary decreases, see Fig. 1 where the projection in the (x, y) plane of thebranches W u

! and W s! until the first and second intersections (respectively) with !, for

different values of µ, are plotted. In fact, for µ > 0.01173615 the projection in the config-uration space of the branch W s

! enters the upper half region x < 0, y > 0. This means

-1.5

-1

-0.5

0

0.5

-1.5 -1 -0.5 0 0.5 1 1.5

y

x

-1.5

-1

-0.5

0

0.5

-1.5 -1 -0.5 0 0.5 1 1.5

y

x

-1.5

-1

-0.5

0

0.5

-1.5 -1 -0.5 0 0.5 1 1.5

y

x

-1.5

-1

-0.5

0

0.5

-1.5 -1 -0.5 0 0.5 1 1.5

y

x

Fig. 1 ! = x = µ ! 1/2. Projection in the configuration space of the invariant manifolds of L3 untilW u

! # !1 and W s! # !2 (superindexes of ! stand for number of intersections) for µ = 0.001 (top left),

µ = 0.005 (top right), µ = 0.02 (bottom left). Bottom right: heteroclinic orbit for µ = 0.014562349014.(Trajectories in W u and W s in red and blue, respectively.) (Color figure online)

123

Jorge Galán Vioque

What is missing

The evolution of the horseshoe periodic orbits is controlledby the existence of symmetric homoclinic orbits to L3(saddle-center).By varying µ it is possible to find multiround homoclinicorbits.The invariant manifolds of the Lyapunov planar familiy ofperiodic orbits born at L1 and L2 is also involved.A systematic investigation of the bifurcation behavior of thehorseshoe orbits is on the way. (connection to elementalfamilies?)A continuation of equlibrium+PO+invariant manifolds andconnections is required.The spatial version is even richer.

Jorge Galán Vioque