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A320478
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.
2
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, 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, 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
OFFSET
0,2
COMMENTS
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.
Note that some websites reduce x(n) mod 2*Pi, but this version does not.
More than the usual number of terms are shown in order to reach an interesting region of terms.
REFERENCES
H. A. Lauwerier, Two-dimensional iterative maps, Chapter 4 of A. V. Holden, ed., Chaos, Princeton, 1986. See Eq. (4.67).
E. N. Lorenz, The Essence of Chaos, Univ. Washington Press, 1993. See p 191.
LINKS
Roderick V. Jensen, Classical chaos, American Scientist 75.2 (1987): 168-181. See Eq. (2), (3).
EXAMPLE
The initial values of x(n) and y(n) are
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, ...
and
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, ...
MAPLE
k:=3; M:=120; x[0]:=1; y[0]:=1;
for n from 1 to M do
y[n]:=y[n-1]+k*evalf(sin(x[n-1]));
x[n]:=x[n-1]+y[n];
od:
[seq(x[n], n=0..M)];
[seq(y[n], n=0..M)];
[seq(round(x[n]), n=0..M)]; # A320478
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 14 2018
STATUS
approved