Abstract
We deal with the orbit determination problem for a class of maps of the cylinder generalizing the Chirikov standard map. The problem consists of determining the initial conditions and other parameters of an orbit from some observations. A solution to this problem goes back to Gauss and leads to the least squares method. Since the observations admit errors, the solution comes with a confidence region describing the uncertainty of the solution itself. We study the behavior of the confidence region in the case of a simultaneous increase in the number of observations and the time span over which they are performed. More precisely, we describe the geometry of the confidence region for solutions in regular zones. We prove an estimate of the trend of the uncertainties in a set of positive measure of the phase space, made of invariant curve. Our result gives an analytical proof of some known numerical evidences.
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References
Albrecht, J.: On the existence of invariant tori in nearly-integrable Hamiltonian systems with finitely differentiable perturbations. Regul. Chaotic Dyn. 12, 281–320 (2007)
Arnol’d, V.I.: Proof of a theorem of A.N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations. Russ. Math. Surv. 18, 9–36 (1963)
Chirikov, B.: A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 263 (1979)
Celletti, A., Di Ruzza, S., Lothka, C., Stefanelli, L.: Nearly-integrable dissipative systems and celestial mechanics. Eur. Phys. J. Spec. Top. 186, 33–66 (2010)
Figueras, J.-L., Haro, À., Luque, A.: On the sharpness of the Rüssmann estimates. Commun. Nonlinear Sci. Numer. Simul. 55, 42–55 (2018)
Gauss, C.F.: Theoria motus corporum coelestium in sectionibus conicis solem ambientium (Theory of the Motion of the Heavenly Bodies Moving About the Sun in Conic Sections). Dover publications (1809/1963)
Gronchi, G.F., Baù, G., Marò, S.: Orbit determination with the two-body integrals: III. Cel. Mech. Dyn. Ast. 123, 105–122 (2015)
González-Enríquez, A., Haro, À., de la Llave, R.: Singularity theory for non-twist KAM tori. Mem. Am. Math. Soc. 227, vi+115 (2014)
Haro, À., Canadell, M., Figueras, J.-L., Luque, A., Mondelo, J.-M.: The Parameterization Method for Invariant Manifolds. Volume 195 of Applied Mathematical Sciences. Springer, Berlin (2016)
Lari, G., Milani, A.: Chaotic orbit determination in the context of the JUICE mission. Planet. Space Sci. 176, 104679 (2019)
Lazutkin, V.F.: Existence of caustics for the billiard problem in a convex domain. Izv. Akad. Nauk SSSR Ser. Mat. 37, 186–216 (1973)
Ma, H., Baù, G., Bracali Cioci, D., Gronchi, G.F.: Preliminary orbits with line-of-sight correction for LEO satellites observed with radar. Cel. Mech. Dyn. Ast. 130, 70 (2018)
Milani, A., Gronchi, G.F.: The Theory of Orbit Determination. Cambridge Univ Press, Cambridge (2010)
Milani, A., Valsecchi, G.B.: The asteroid identification problem II: target plane confidence boundaries. Icarus 140, 408–423 (1999)
Pöschel, J.: Integrability of Hamiltonian systems on Cantor sets. Commun. Pure Appl. Math. 35, 653–696 (1982)
Pöschel, J.: A lecture on the classical KAM theory, Katok, Anatole (ed.) et al., Smooth ergodic theory and its applications (Seattle, WA, 1999). Providence, RI: Amer. Math. Soc. (AMS). Proc. Symp. Pure Math. 69, 707–732 (2001)
Rüssman, H.: On optimal estimates for the solutions of linear difference equations on the circle. Cel. Mech. 14, 33–37 (1976)
Siegel, C.L., Moser, J.K.: Lectures on Celestial Mechanics. Springer, Berlin (1971)
Serra, D., Spoto, F., Milani, A.: A multi-arc approach for chaotic orbit determination problems. Cel. Mech. Dyn. Ast. 130, 75 (2018)
Shang, Z.: A note on the KAM theorem for symplectic mappings. J. Dyn. Differ. Eq. 12, 357–383 (2000)
Spoto, F., Milani, A.: Shadowing Lemma and chaotic orbit determination. Cel. Mech. Dyn. Astron. 124, 295–309 (2016)
Acknowledgements
This problem was proposed to me by Andrea Milani. This result and possible further developments are dedicated to his memory. I would like to thank the unknown referees for several valuable advice that significantly improved the final version of the paper.
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This work was supported by the National Group of Mathematical Physics (GNFM-INdAM) through the project “Orbit Determination: from order to chaos” (Progetto Giovani 2019).
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Marò, S. Orbit determination for standard-like maps: asymptotic expansion of the confidence region in regular zones. Celest Mech Dyn Astr 132, 40 (2020). https://doi.org/10.1007/s10569-020-09980-6
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DOI: https://doi.org/10.1007/s10569-020-09980-6