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.

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

Table of n, a(n) for n=0..103.

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

Cf. A320472-A320480.

Sequence in context: A332507 A242617 A158349 * A364881 A225302 A079384

Adjacent sequences: A320475 A320476 A320477 * A320479 A320480 A320481

Keyword

nonn

Author

N. J. A. Sloane, Oct 14 2018

Status

approved