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%I #15 Oct 15 2018 05:07:07
%S 1,5,5,3,1,3,5,5,1,0,0,1,2,6,10,12,12,11,8,7,10,11,9,8,10,10,8,9,11,
%T 10,8,9,12,11,9,8,10,10,8,9,11,10,7,6,6,5,2,1,4,4,2,2,5,4,1,0,1,5,6,8,
%U 13,18,21,26,35,42,47,52,60,67,72,78,86,91,96,104,112,117,121,128,137,145,154,163,172,183,196,211,225,238,247,259,274,287,297,310,326,339,350,358,368,377,386,397
%N a(n) = round(x(n)), where (x(n),y(n)) are defined by the Chirikov "standard map" y(n) = y(n-1) + 3*sin(x(n-1)), x(n) = x(n-1) + y(n), with x(0)=y(0)=1.
%C The Chirikov map is an example of a nonlinear dynamical system which can exhibit chaotic behavior. Most such maps do not easily lead to integer sequences, but this map does.
%C Note that some websites reduce x(n) mod 2*Pi, but this version does not.
%C More than the usual number of terms are shown in order to reach an interesting region of terms.
%D H. A. Lauwerier, Two-dimensional iterative maps, Chapter 4 of A. V. Holden, ed., Chaos, Princeton, 1986. See Eq. (4.67).
%D E. N. Lorenz, The Essence of Chaos, Univ. Washington Press, 1993. See p 191.
%H Roderick V. Jensen, <a href="https://www.jstor.org/stable/27854538">Classical chaos</a>, American Scientist 75.2 (1987): 168-181. See Eq. (2), (3).
%e The initial values of x(n) and y(n) are
%e 1, 4.524412954, 5.101672501, 2.903388389, 1.412978211, 2.885285674, 5.118122864, 4.594521097, 1.091734472, 0.251231092, 0.1565174874, 0.5294415308, 2.417519808, 6.292921635, 10.19753198, 12.00781312, 12.22820384, 11.45331980, 7.987282478, ...
%e and
%e 1, 3.524412954, 0.577259547, -2.198284112, -1.490410178, 1.472307463, 2.232837190, -0.523601767, -3.502786625, -0.840503380, -0.0947136046, 0.3729240434, 1.888078277, 3.875401827, 3.904610349, 1.810281141, 0.220390715, ...
%p k:=3; M:=120; x[0]:=1; y[0]:=1;
%p for n from 1 to M do
%p y[n]:=y[n-1]+k*evalf(sin(x[n-1]));
%p x[n]:=x[n-1]+y[n];
%p od:
%p [seq(x[n],n=0..M)];
%p [seq(y[n],n=0..M)];
%p [seq(round(x[n]),n=0..M)]; # A320478
%Y Cf. A320472-A320480.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Oct 14 2018
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