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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 (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..67.

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

Cf. A320472-A320480.

Sequence in context: A016718 A165361 A197136 * A106626 A280191 A222703

Adjacent sequences:  A320472 A320473 A320474 * A320476 A320477 A320478

KEYWORD

sign

AUTHOR

N. J. A. Sloane, Oct 14 2018

STATUS

approved

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Last modified August 2 18:40 EDT 2021. Contains 346428 sequences. (Running on oeis4.)