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35949021570
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``Chaotic'' is defined by the property of nearby trajectories to diverge.
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(1981)
Rev. Mod. Phys.
, vol.53
, pp. 655
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Ott, E.1
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5
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0000954445
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A method of distinguishing stochastic and (infinitely differentiable) deterministic time series directly from the power spectrum is discussed by
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(1987)
Phys. Rev. A
, vol.35
, pp. 2276
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Sigeti, D.1
Horsthemke, W.2
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6
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84926837044
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However, involves the asymptotic high-frequency behavior and this may not be available in a discrete time series. Also, does not measure the number of degrees of freedom in a deterministic system.
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13
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84926858778
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The correlation dimension was discussed independently by F. Takens, Invariants Related to Dimension and Entropy, in Atas do 13degr. Colóqkio Brasiliero de Matemática, Rio de Janeiro, 1983 (unpublished).
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16
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84926793323
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J. Theiler, Ph.D. thesis, California Institute of Technology, 1987.
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17
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84926858777
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F. Takens, in Detecting Strange Attractors in Turbulence, Vol. 898 of Lecture Notes in Mathematics, edited by D. A. Rand and L.-S. Young (Springer-Verlag, Berlin, 1981).
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19
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0018999905
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A general discussion of related problems from a computer science viewpoint can be found in
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(1980)
Commun. ACM
, vol.23
, pp. 214
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Bentley, J.L.1
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20
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84926837042
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We are indebted to J. D. Farmer for this reference.
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21
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84926858776
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This assumes the Linf or ``maximum'' metric. For the L1 or ``taxicab'' metric, the appropriate inequality is r <= 2mr0; for the L2 or Euclidean metric, is r <= 2 sqrt m r0.
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84926815309
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In the ``lexicographic'' ordering, we say that a vec < b vec, where in coordinate notation a vec =(a1, a2,..., am) and b vec =(b1, b2,..., bm), if and only if ak< bk for some k <= m, and ai=biinssA i < k.
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23
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84926858775
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On our IBM PC, we find τdist=(0.5 +0.2m) msec. In the Takens method, we have also to take a logarithm of all distances less than r0; these cost 4.3 msec each (with an 8087 floating point co-processor). Also τsearchapprox (1.5+0.6m) msec and τsort=(0.44+0.09m) msec, with some leveling off at large m. Finally, τread=6.7 msec. There is a small memory compiler option which cuts these times in half, but can only be used for N<5000.
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24
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84926815308
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Takens (in Ref. 17) says the error bar on nu will scale as 1/ Dstar, where Dstar is the number of distances less than r0.
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25
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84926858774
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Properly we should write B approx (2R/r0)d, where d is the ``capacity'' of the set. Though capacity and correlation dimension are not precisely the same thing, our approximations do not distinguish between them.
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84926837041
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What we really assume is that points are distributed uniformly among those boxes which cover the attractor; this presumes not only that the distribution of points is uniform over the attractor, but that the intersection of the attractor with boxes is uniform—in fact, there are often a lot of ``clipped edges.'' For details of this effect in another context, see W. E. Caswell and J. A. Yorke, in Dimensions and Entropies in Chaotic Systems, Vol. 32 of Springer Series in Synergetics, edited by G. Mayer-Kess (Springer-Verlag, Berlin, 1986), p. 123.
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84926858773
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The source code, further documentation, and executable files which run on an IBM PC (or compatible) are available from the author.
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84926815307
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In our Grassberger-Procaccia routine, we use 8N +4xr0 bytes, where x is the ``expansion factor'' which is multiplied by the floating point input before discretizing into integers. In our Takens maximum likelihood routine, we use 14N bytes since the floating point input is stored in double precision.
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30
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84926815306
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The mapping is xi+1=yi+1-axi2;
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31
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84926837040
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yi+1=bxi. Following Hénon, we use a=1.4, b=0.3.
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84926837039
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For low-dimensional attractor data, we find that the search time increases so slowly with m that O(N) distances can always be computed faster with the box-assisted algorithm than with the standard algorithm. But even in these cases, we find that the ``prism-assisted'' variant provides still further improvement.
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