Shadowing Lemma and chaotic orbit determination Federica Spoto 1 , A. Milani 2 fspoto@oca.eu ote d’Azur, CNRS 1 , Nice Laboratoire Lagrange, Observatoire de la Cˆ University of Pisa 2 , Pisa Dynamics and chaos in astronomy and physics 2016 September 17-24, Luchon
Chaotic orbit determination What happens if we need to have an accurate quantitative knowledge of chaotic orbits? “In fact because of the exponential variety of trajectories which exists, the rotation state at the midpoint of the interval covered by 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 the time span covered by the observations. ” Wisdom, J. Urey Prize Lecture: Chaotic Dynamics in the Solar System Icarus 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 discrete dynamical system, defined on a 2-dimensional torus, which has both ordered and chaotic orbits. � x k + 1 = x k + y k + 1 S µ ( x 0 , y 0 ) = y k + 1 = y k − µ sin ( x k ) . Standard map (mu=0.5) The system has more regular 0.2 0.15 orbits for small µ , and more 0.1 chaotic orbits for large µ . 0.05 We choose an intermediate y 0 value µ = 0 . 5, in such a way −0.05 −0.1 that both ordered and chaotic −0.15 orbits are present. −0.2 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 x F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)
Standard map linearization The least square parameter estimation process can be performed by an explicit formula . • Linearized map: � ∂ x k + 1 � 1 − µ cos ( x k ) ∂ x k + 1 � � 1 x k y k DS = = ∂ y k + 1 ∂ y k + 1 − µ cos ( x k ) 1 x k y k • Linearized state transition matrix: A k = ∂ ( x k , y k ) ∂ ( x 0 , y 0 ); A k + 1 = DS A k ; A 0 = I • Variational equation: ∂ ( x k + 1 , y k + 1 ) DS ∂ ( x k , y k ) + ∂ S = ∂µ ∂µ ∂µ � − sin ( x k ) � DS ∂ ( x k , y k ) = + ∂µ − sin ( x k ) 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 variables with mean x k ( y k , respectively) and standard deviation σ . Residuals • The residuals contain two components: a random one for the observation error, and a systematic one because the true value µ 0 is not the same as the current guess. � ξ k = x k ( µ 0 , σ ) − x k ( µ 1 ) ¯ ξ k = y k ( µ 0 , σ ) − y k ( µ 1 ) , 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 normal equations: � ξ k n n � � B T � B T C = k B k ; D = − ¯ k ξ k k = − n k = − n ∂ ( ξ k , ¯ ξ k ) � A k | ∂ ( x k , y k ) � B k = ∂ ( x 0 , y 0 , µ ) = − ∂µ • An iteration of differential corrections is a correction Γ D 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 the residuals consistently with the probabilistic model. F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)
Shadowing Lemma: δ -pseudotrajectory δ -pseudotrajectory A δ -pseudotrajectory is a sequence of points ( x k , y k ) connected by an approximation of the map Φ , with error < δ at each step: | Φ( x k , y k ) − ( x k + 1 , y k + 1 ) | < δ F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)
Shadowing Lemma: ε -shadowing ε -shadowing The orbit with initial conditions ( x , y ) ( ε , Φ )-shadows a δ -pseudotrajectory ( x k , y k ) if: | Φ k ( x , y ) − ( x k , y k ) | < ε for every k . F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)
Shadowing Lemma Shadowing Lemma If Λ is an hyperbolic set for a diffeomorphism Φ , then there exists 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 W that ε -shadows the δ -pseudotrajectory. • An hyperbolic set is (rougly) an invariant set with every orbit having a positive and a negative Lyapounov exponent. • There is an L > 0, function of the Lyapounov exponents, such that δ < ε/ L . F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)
Shadowing Lemma and least squares solution I To connect orbit determination and the Shadowing Lemma, we need to show first that the observations x k ( µ 0 , σ ) , y k ( µ 0 , σ ) are a δ -pseudotrajectory for the dynamical system S µ ∗ : • µ ∗ is the value of the dynamical parameter found from the least squares solution √ • δ = 2 | µ 0 − µ ∗ | + K σ Delta−pseudotrajectory for the standard map 2 (xdelta,ydelta) Example of a (x,y) 1.5 δ -pseudotrajectory. 1 0.5 Initial conditions : 0 y x 0 = 3, y 0 = 0, µ 0 = 0 . 5. −0.5 Options : −1 δµ = 10 − 1 , σ = 10 − 3 . −1.5 −2 0 1 2 3 4 5 6 x F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)
Shadowing Lemma and least squares solution II • We select ε > K σ : K is a number such that, at convergence, no larger norm | ( ξ k , ¯ ξ k ) | is found among the residuals for − n ≤ k ≤ n • The orbit with initial conditions ( x ∗ , y ∗ ) ε -shadows the δ -pseudotrajectory formed by the observations ( x k , y k ) . Summary • The observations are a pseudotrajectory because of errors and systematics due to imperfect knowledge of the dynamics. • The least squares solution is the shadowing of the observations. • The Shadowing Lemma is a minimization of the infinite dimensional space of all orbits , while the orbit determination is a minimization of the norm of a finite number of residuals . F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)
Computability Horizon Determinant and eigenvalues of the state transition matrix 30 Max. eigenvalue 20 10 log 0 −10 −20 Min. eigenvalue −30 −300 −200 −100 0 100 200 300 # iterations • Initial conditions: x 0 = 3, y 0 = 0, µ 0 = 0 . 5 • Positive Lyapounov exponent is χ ≃ 0 . 091, and the Lyapounov time is t L = 1 /χ ≃ 11. • The numerical instability at k ≃ 200 occurs because exp ( 200 / t L ) ∼ 10 8 . • The inversion of the normal matrix C fails. F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)
Meaning of the computability horizon Determinant and eigenvalues of the state transition matrix 80 60 Max. eigenvalue 40 20 log 0 −20 Min. eigenvalue −40 −60 −800 −600 −400 −200 0 200 400 600 800 # iterations • 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 a δ -pseudotrajectory. F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)
Results: with and without dynamical parameter Uncertainty 2 parameters Uncertainty 3 parameters −20 −20 −25 σ µ −30 −30 σ x −35 σ y −40 −40 log log −45 −50 σ y σ x −50 −60 −55 −60 −70 −65 −70 −80 0 100 200 300 400 500 600 700 800 0 100 200 300 400 500 600 # iterations # iterations • Least squares solution for up to 600 / 700 iterates in quadruple precision, by using a progressive method. • If the solutions has a fixed µ = µ 0 , the improvement in accuracy is exponential in k . • If we solve for 2 initial conditions and µ , the improvements is not exponential in at least two variables (including µ ). F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)
Loss of accuracy when adding a dynamical parameter Standard map (mu=0.5) 12 10 8 dcsi/dmu 6 log10 4 2 0 −2 −4 −300 −200 −100 0 100 200 300 No. iterations • The much lower accuracy in the determination of µ and at least 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. F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)
Results: chaotic orbit, power law improvement Decrease uncertainty and fit (3 par: x, y) 0 −10 −20 σ µ −30 σ x log σ y −40 −50 −60 −70 0 1 2 3 4 5 6 7 log(# iterations) • Power law accuracy improvement with the number n of iterations like n a 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. F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)
Results: ordered case, power law improvement • Initial conditions: x 0 = 2, y 0 = 0, µ 0 = 0 . 5 • The computability horizon disappears. • The Lyapounov exponent could be zero. F. Spoto Chaotic orbit determination Luchon (2016 Sept. 19)
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