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Orbit determination is possible for a chaotic orbit of a dynamical system, given a
finite set of observations, provided the initial conditions are at the central time. The Shadowing
Lemma (Anosov 1967; Bowen in J Differ Equ 18:333–356, 1975) can be seen as a way to
connect the orbit obtained using the observations with a real trajectory.Anorbit is a shadowing
of the trajectory if it stays close to the real trajectory for some amount of time. In a simple
discrete model, the standard map, we tackle the problem of chaotic orbit determination when
observations extend beyond the predictability horizon. If the orbit is hyperbolic, a shadowing
orbit is computed by the least squares orbit determination. We test both the convergence
of the orbit determination iterative procedure and the behaviour of the uncertainties as a
function of the maximum number of map iterations observed. When the initial conditions
belong to a chaotic orbit, the orbit determination is made impossible by numerical instability
beyond a computability horizon, which can be approximately predicted by a simple formula.
Moreover, the uncertainty of the results is sharply increased if a dynamical parameter is
added to the initial conditions as parameter to be estimated. The Shadowing Lemma does
not dictate what the asymptotic behaviour of the uncertainties should be. These phenomena
have significant implications, which remain to be studied, in practical problems of orbit
determination involving chaos, such as the chaotic rotation state of a celestial body and a
chaotic orbit of a planet-crossing asteroid undergoing many close approaches
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