A320480 - OEIS

A320480

a(n) = floor(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, 4, 5, 2, 1, 2, 5, 4, 1, 0, 0, 0, 2, 6, 10, 12, 12, 11, 7, 7, 9, 10, 9, 8, 9, 10, 8, 9, 10, 9, 7, 8, 11, 11, 8, 7, 10, 10, 8, 8, 11, 10, 7, 6, 6, 5, 1, 1, 3, 4, 2, 2, 4, 4, 0, 0, 1, 5, 6, 8, 12, 18, 21, 26, 34, 42, 46, 52, 59, 67, 71, 77, 86, 91, 95, 103, 111, 116, 120, 127, 137, 144, 153, 163, 172, 182, 195, 210, 225

(list; graph; refs; listen; history; text; internal format)

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..88.

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(floor(x[n]), n=0..M)]; # A320480

CROSSREFS

Cf. A320472-A320479.

Sequence in context: A274615 A258895 A156890 * A163531 A336199 A267095

Adjacent sequences: A320477 A320478 A320479 * A320481 A320482 A320483

KEYWORD

sign

AUTHOR

N. J. A. Sloane, Oct 14 2018

STATUS

approved