[2.13] Can computers simulate chaos?
Strictly speaking, chaos cannot occur on computers because they deal with fi
nite sets of numbers. Thus the initial condition is always precisely known,
and computer experiments are perfectly predictable, in principle. In particu
lar because of the finite size, every trajectory computed will eventually ha
ve to repeat (an thus be eventually periodic). On the other hand, computers
can effectively simulate chaotic behavior for quite long times (just so long
as the discreteness is not noticeable). In particular if one uses floating
point numbers in double precision to iterate a map on the unit square, then
there are about 10^28 different points in the phase space, and one would exp
ect the "typical" chaotic orbit to have a period of about 10^14 (this square
root of the number of points estimate is given by Rannou for random diffeom
orphisms and does not really apply to floating point operations, but nonethe
less the period should be a big number). See, e.g.,
Earn, D. J. D. and S. Tremaine, "Exact Numerical Studies of Hamiltonian Maps
: Iterating without Roundoff Error," Physica D 56, 1-22 (1992).
Binder, P. M. and R. V. Jensen, "Simulating Chaotic Behavior with Finite Sta
te Machines," Phys. Rev. 34A, 4460-3 (1986).
Rannou, F., "Numerical Study of Discrete Plane Area-Preserving Mappings," As
tron. and Astrophys. 31, 289-301 (1974).
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在細雨的午後 書頁裡悉哩哩地傳來 " 週期3 = ? "
然而我知道 當我正在日耳曼深處的黑森林
繼續發掘海森堡未曾做過的夢時 康德的諾言早已遠離.........
遠來的傳教士靜靜地看著山澗不斷反覆疊代自己的 過去 現在 和 未來
於是僅以 一顆量子渾沌
一本符號動力學 祝那發生在週一下午的新生
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