TY - JOUR
T1 - Shadowing by computable chaotic orbits
AU - Palmore, Julian I.
AU - McCauley, Joseph L.
N1 - Funding Information:
Julian Palmore thanks T. Riste and the Institute for Energy Technology for hospitality during his visit to Kjeller, Norway in January, 1986, to pursue research in chaotic dynamics. J.L. McCauley is grateful to T. Riste and the Institute for Energy Technology for guestfriendsbip during a free semester and thanks P. Cvitanovic for a remark. This work was supported in part by NORDITA and The American— Scandinavian Foundation.
Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.
PY - 1987/6/29
Y1 - 1987/6/29
N2 - We report new results on the shadowing of computable nonperiodic pseudo-orbits by computable chaotic orbits. While ordinary machine truncation/roundoff decisions can produce only periodic pseudo-orbits, we show how nonperiodic pseudo-orbits of chaotic maps can be generated by using for the truncation decision an algorithm for a computable irrational number. The resulting nonperiodic pseudo-orbits are shadowed by unique chaotic orbits of the dynamical system. We illustrate this by constructing examples of nonperiodic pseudo-orbits along with their unique chaotic shadowing orbits for a hyperbolic system. We conclude that the β-shadowing lemma without additional hypotheses provides no information for inferring chaotic attractor statistics from pseudo-orbit statistics in computation.
AB - We report new results on the shadowing of computable nonperiodic pseudo-orbits by computable chaotic orbits. While ordinary machine truncation/roundoff decisions can produce only periodic pseudo-orbits, we show how nonperiodic pseudo-orbits of chaotic maps can be generated by using for the truncation decision an algorithm for a computable irrational number. The resulting nonperiodic pseudo-orbits are shadowed by unique chaotic orbits of the dynamical system. We illustrate this by constructing examples of nonperiodic pseudo-orbits along with their unique chaotic shadowing orbits for a hyperbolic system. We conclude that the β-shadowing lemma without additional hypotheses provides no information for inferring chaotic attractor statistics from pseudo-orbit statistics in computation.
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U2 - 10.1016/0375-9601(87)90736-5
DO - 10.1016/0375-9601(87)90736-5
M3 - Article
AN - SCOPUS:0001475851
VL - 122
SP - 399
EP - 402
JO - Physics Letters, Section A: General, Atomic and Solid State Physics
JF - Physics Letters, Section A: General, Atomic and Solid State Physics
SN - 0375-9601
IS - 8
ER -