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A320475
a(n) = round(x(n)), where (x(n),y(n)) are defined by the Chirikov "standard map" y(n) = y(n-1) + 2*sin(x(n-1)), x(n) = x(n-1) + y(n), with x(0)=y(0)=1.
2
1, 4, 5, 5, 4, 1, 0, -1, -5, -6, -6, -5, -3, -1, 0, -1, -2, -5, -6, -7, -9, -12, -14, -18, -19, -23, -25, -25, -27, -31, -33, -36, -38, -41, -43, -44, -46, -49, -51, -55, -57, -58, -62, -64, -68, -70, -74, -76, -79, -81, -82, -84, -87, -89, -92, -94, -94, -95, -97, -100, -100, -101, -101, -103, -106, -107, -109, -112
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.
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, 3.682941970, 5.335298253, 5.363285286, 3.800190225, 1.013078481, -0.077102958, -1.321337570, -4.503664553, -5.729399487, -5.903312461, -5.335620669, -3.143935907, ...
and
1, 2.682941970, 1.652356283, 0.027987033, -1.563095061, -2.787111744, -1.090181439, -1.244234612, -3.182326983, -1.225734934, -0.173912974, 0.5676917924, 2.191684762, ...
MAPLE
k:=2; 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)]; # A320475
CROSSREFS
KEYWORD
sign
AUTHOR
N. J. A. Sloane, Oct 14 2018
STATUS
approved