A320474 a(n) = floor(x(n)), where (x(n),y(n)) are defined by the Chirikov "standard map" y(n) = y(n-1) + sin(x(n-1)), x(n) = x(n-1) + y(n), with x(0)=y(0)=1.

1, 2, 4, 6, 7, 9, 11, 12, 13, 15, 17, 18, 20, 22, 24, 25, 26, 28, 30, 31, 32, 34, 36, 37, 38, 40, 42, 43, 45, 47, 49, 50, 51, 53, 55, 56, 57, 59, 61, 62, 63, 65, 67, 68, 70, 71, 74, 75, 76, 78, 80, 81, 83, 85, 87, 88, 89, 91, 93, 94, 95, 96, 99, 100, 101, 103, 105, 106, 108, 110, 112, 113, 114, 116, 118, 119, 120, 122

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

Table of n, a(n) for n=0..77.

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, 2.841470985, 4.978578409, 6.150905537, 7.191338328, 9.020139937, 11.24262759, 12.49547800, 13.67749516, 15.75571048, 17.78619673, 18.94269021, 20.19218340, 22.41575881, 24.22736856, ...
and
1, 1.841470985, 2.137107424, 1.172327128, 1.040432791, 1.828801609, 2.222487650, 1.252850411, 1.182017163, 2.078215324, 2.030486252, 1.156493480, 1.249493186, 2.223575410, 1.811609751, ...

Maple

k:=1; 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(floor(x[n]), n=0..M)]; # A320494

Crossrefs

Cf. A320472-A320480.

Sequence in context: A356754 A187974 A193715 * A051533 A186151 A184732

Adjacent sequences: A320471 A320472 A320473 * A320475 A320476 A320477

Keyword

nonn

Author

N. J. A. Sloane, Oct 14 2018

Status

approved