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A320472 a(n) = round(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. 6
1, 3, 5, 6, 7, 9, 11, 12, 14, 16, 18, 19, 20, 22, 24, 25, 26, 28, 30, 31, 32, 34, 36, 38, 39, 40, 43, 44, 45, 47, 49, 50, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 77, 79, 81, 82, 83, 85, 87, 88, 89, 91, 93, 94, 95, 97, 99, 100, 101, 103, 106, 107, 108, 110, 112, 113, 114, 117, 118, 119, 121, 123 (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.

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(round(x[n]), n=0..M)]; # A320492

CROSSREFS

Cf. A320473-A320480.

Sequence in context: A286908 A284852 A285627 * A064802 A233570 A181766

Adjacent sequences:  A320469 A320470 A320471 * A320473 A320474 A320475

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 14 2018

STATUS

approved

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Last modified October 16 01:27 EDT 2019. Contains 328038 sequences. (Running on oeis4.)