shadowing lemma and chaotic orbit determination · 2016. 9. 25. · f. spoto chaotic orbit...

Shadowing Lemmaand

chaotic orbit determination

Federica Spoto1, A. Milani2

[email protected]

Laboratoire Lagrange, Observatoire de la Cote d’Azur, CNRS1, NiceUniversity of Pisa2, Pisa

Dynamics and chaos in astronomy and physics2016 September 17-24, Luchon

Chaotic orbit determination

What happens if we need to have an accurate quantitativeknowledge of chaotic orbits?

“In fact because of the exponential variety of trajectories whichexists, the rotation state at the midpoint of the interval coveredby the observations, and the principal moments of inertia, are

determined with exponential accuracy.Thus the knowledge gained from measurements on a

chaotic dynamical system grows exponentially with thetime span covered by the observations.”

Wisdom, J.Urey Prize Lecture:

Chaotic Dynamics in the Solar SystemIcarus 72, 241-275 (1987)

F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)

Standard Map: Definition

The standard map of the pendulum is a conservative discretedynamical system, defined on a 2-dimensional torus, which hasboth ordered and chaotic orbits.

Sµ(x0, y0) =

{xk+1 = xk + yk+1yk+1 = yk − µ sin(xk ).

2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2Standard map (mu=0.5)

x

y

The system has more regularorbits for small µ, and morechaotic orbits for large µ.We choose an intermediatevalue µ = 0.5, in such a waythat both ordered and chaoticorbits are present.


Standard map linearization

The least square parameter estimation process can beperformed by an explicit formula.

• Linearized map:

DS =

(∂xk+1

xk

∂xk+1yk

∂yk+1xk

∂yk+1yk

)=

(1− µ cos(xk ) 1−µ cos(xk ) 1

)• Linearized state transition matrix:

Ak =∂(xk , yk )

∂(x0, y0); Ak+1 = DS Ak ; A0 = I

• Variational equation:

∂(xk+1, yk+1)

∂µ= DS

∂(xk , yk )

∂µ+∂S∂µ

= DS∂(xk , yk )

∂µ+

(− sin(xk )− sin(xk )

)F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)

Standard map orbit determination I

Observations process

• Both coordinates x and y are observed at each iteration,and the observations are Gaussian random variableswith mean xk (yk , respectively) and standard deviation σ.

Residuals

• The residuals contain two components: a random one forthe observation error, and a systematic one because thetrue value µ0 is not the same as the current guess.{

ξk = xk (µ0, σ)− xk (µ1)ξk = yk (µ0, σ)− yk (µ1),

with k = −n, . . . ,n.F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)

Standard map orbit determination II

• The least squares fit is obtained from the normalequations:

C =n∑

k=−n

BTk Bk ; D = −

n∑k=−n

BTk

(ξkξk

)

Bk =∂(ξk , ξk )

∂(x0, y0, µ)= −

(Ak |

∂(xk , yk )

∂µ

)• An iteration of differential corrections is a correction ΓD

obtained from the covariance matrix Γ = C−1.• At convergence of the iterations to the least squares

solution (x∗, y∗, µ∗), weights should be assigned to theresiduals consistently with the probabilistic model.


Shadowing Lemma: δ-pseudotrajectory

δ-pseudotrajectory

A δ-pseudotrajectory is a sequence of points (xk , yk ) connectedby an approximation of the map Φ, with error < δ at each step:

|Φ(xk , yk )− (xk+1, yk+1)| < δ


Shadowing Lemma: ε-shadowing

ε-shadowing

The orbit with initial conditions (x , y) (ε, Φ)-shadows aδ-pseudotrajectory (xk , yk ) if:

|Φk (x , y)− (xk , yk )| < ε

for every k .


Shadowing Lemma

Shadowing LemmaIf Λ is an hyperbolic set for a diffeomorphism Φ, then thereexists a neighborhood W of Λ such that

for every ε > 0 there exists δ > 0

such that

for every δ-pseudotrajectory in W there exists a point in Wthat ε-shadows the δ-pseudotrajectory.

• An hyperbolic set is (rougly) an invariant set with everyorbit having a positive and a negative Lyapounov exponent.

• There is an L > 0, function of the Lyapounov exponents,such that δ < ε/L.


Shadowing Lemma and least squares solution I

To connect orbit determination and the Shadowing Lemma, weneed to show first that the observations xk (µ0, σ), yk (µ0, σ) area δ-pseudotrajectory for the dynamical system Sµ∗ :• µ∗ is the value of the dynamical parameter found from the

least squares solution• δ =

√2|µ0 − µ∗|+ Kσ

0 1 2 3 4 5 6−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Delta−pseudotrajectory for the standard map

x

y

(xdelta,ydelta)(x,y) Example of a

δ-pseudotrajectory.

Initial conditions:x0 = 3, y0 = 0, µ0 = 0.5.

Options:δµ = 10−1, σ = 10−3.


Shadowing Lemma and least squares solution II

• We select ε > Kσ: K is a number such that, atconvergence, no larger norm |(ξk , ξk )| is found among theresiduals for −n ≤ k ≤ n

• The orbit with initial conditions (x∗,y∗) ε-shadows theδ-pseudotrajectory formed by the observations (xk , yk ).

Summary

• The observations are a pseudotrajectory because oferrors and systematics due to imperfect knowledge of thedynamics.

• The least squares solution is the shadowing of theobservations.

• The Shadowing Lemma is a minimization of the infinitedimensional space of all orbits, while the orbitdetermination is a minimization of the norm of a finitenumber of residuals.


Computability Horizon

−300 −200 −100 0 100 200 300−30

−20

−10

0

10

20

30Determinant and eigenvalues of the state transition matrix

# iterations

log

Max. eigenvalue

Min. eigenvalue

• Initial conditions: x0 = 3, y0 = 0, µ0 = 0.5• Positive Lyapounov exponent is χ ' 0.091, and the

Lyapounov time is tL = 1/χ ' 11.• The numerical instability at k ' 200 occurs because

exp(200/tL) ∼ 108.• The inversion of the normal matrix C fails.


Meaning of the computability horizon

−800 −600 −400 −200 0 200 400 600 800−60

−40

−20

0

20

40

60

80Determinant and eigenvalues of the state transition matrix

# iterations

log

Max. eigenvalue

Min. eigenvalue

• The computability horizon is ' 600 iterations of S.• The computability horizon is an absolute barrier to the

determination of a least squares orbit.• We need to admit that we can only solve for aδ-pseudotrajectory.


Results: with and without dynamical parameter

0 100 200 300 400 500 600 700 800−80

−70

−60

−50

−40

−30

−20Uncertainty 2 parameters

# iterations

log

σ x

σ y

0 100 200 300 400 500 600−70

−65

−60

−55

−50

−45

−40

−35

−30

−25

−20Uncertainty 3 parameters

# iterations

log

σ µ

σ x

σ y

• Least squares solution for up to 600/700 iterates inquadruple precision, by using a progressive method.

• If the solutions has a fixed µ = µ0, the improvement inaccuracy is exponential in k .

• If we solve for 2 initial conditions and µ, the improvementsis not exponential in at least two variables (including µ).


Loss of accuracy when adding a dynamical parameter

−300 −200 −100 0 100 200 300−4

−2

0

2

4

6

8

10

12Standard map (mu=0.5)

No. iterations

log10

dcsi/dmu

• The much lower accuracy in the determination of µ and atleast one initial condition is not due to lack of sensitivity.

• Correlations grow.• Orbit determination is degraded by aliasing.• This is a finite-time analog of the shadowing lemma.


Results: chaotic orbit, power law improvement

0 1 2 3 4 5 6 7−70

−60

−50

−40

−30

−20

−10

0Decrease uncertainty and fit (3 par: x, y)

log(# iterations)

log

σ x

σ µ

σ y

• Power law accuracy improvement with the number n ofiterations like na with a = −0.675 for µ, a = −0.833 for x ,while for y a power law is not appropriate.

• The slopes are sensitive to the initial conditions.


Results: ordered case, power law improvement

• Initial conditions: x0 = 2, y0 = 0, µ0 = 0.5• The computability horizon disappears.• The Lyapounov exponent could be zero.


Results: ordered case, gaussian statistics

0 1 2 3 4 5 6 7 8 9−29

−28

−27

−26

−25

−24

−23Decrease uncertainty and fit (3 par: x, y)

log(# iterations)

log σ x

σ y

σ µ

• The difference between the case with 2 parameters andthe one with 3 disappears.

• In a log-log plot we find an accuracy improvement na witha = −0.504,−0.504,−0.488 for µ, x , y respectively.

• We conjectured that the exponent is actually −0.5.


Conclusions

Orbit determination is possible only within thecomputability horizon.

If a parameter is estimated, the least squares solution is afinite time span shadowing of the observations

Chaotic orbit: the precision of the solution can grow only as apower law na with −1 < a < 0.

Ordered orbit: the computability horizon is not a problem andthe precision improvement is na with a ' −1/2.

How soon will we find practical problems of dynamicalastronomy in which we need to use these concepts and

preliminary results?


Example from impact monitoring

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

ξ (unit R⊕

)

ζ (

unit R

⊕)

−4 −3 −2 −1 0 1 2 3 4

−2

−1

0

1

2

3

4

5

xi, Earth radii

zeta

, E

art

h r

adii

• The twin Virtual Impactors (VI) (connected patches ofinitial conditions colliding with Earth) for asteroid (101955)Bennu in year 2182.

• Orbits compatible with the observations (up to 2011) and aYarkovsky model.

• Vertical axis is along the projection of Earth’s velocity ontothe TP.


Example from impact monitoring

0 5 10 15 20 25 30 35 40 45−8

−6

−4

−2

0

2

4

6

8

10

12

shower no

log10(a

bs(e

igenvalu

es))

Eigenvalues of the state transition matrix

• State transition matrix for the integral flow of the equationsof motion for (101955) Bennu, after the ca of 2182.

• The eigenvalues are 6, 2 real and 4 complex, always incouples λ, 1/λ.


Determination of Yarkovsky effect

−6 −5 −4 −3 −2 −1 0 1 2

x 106

1

2

3

4

5

6x 10

−6

ζ2185

− km

Pro

b. D

ensity −

km

−1

10−1

100

101

102

103

Keyhole

Wid

th −

km

• Prediction on the Target Plane of an encounter in the 2185with a Yarkovsky dynamical parameter estimated, andwithout it.

• The keyholes for impact in the year 2185 to 2196 are all inthe possible range of values with Yarkovsky.


shadowing lemma and chaotic orbit determination · 2016. 9. 25. · f. spoto chaotic orbit...

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