Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Shadowing Lemma and Chaotic Orbit Determination   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. 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 $n$ 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 uncertainty of the dynamical parameter decreases like $n^a$ with
$a<0$ but not large (of the order of unity). If only the initial conditions are
estimated, their uncertainty decreases exponentially with $n$. If they belong
to a non-chaotic orbit the computational horizon is much larger, if it exists
at all, and the decrease of the uncertainty is polynomial in all parameters,
like $n^a$ with $a\simeq 1/2$. 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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The categories of this paper are: plasm-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: eess
0
The categories of this paper are: hep-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int
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The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: patt-sol
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: cs
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys