A320477 a(n) = floor(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.

1, 3, 5, 5, 3, 1, -1, -2, -5, -6, -6, -6, -4, -1, -1, -1, -2, -5, -7, -7, -9, -12, -14, -18, -20, -23, -25, -26, -28, -31, -33, -37, -39, -42, -44, -45, -46, -50, -52, -55, -57, -59, -62, -65, -69, -71, -75, -77, -80, -82, -83, -84, -88, -89, -93, -94, -95

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..56.

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(floor(x[n]), n=0..M)]; # A320477

Crossrefs

Cf. A320472-A320480.

Sequence in context: A078063 A373304 A019944 * A110551 A141334 A199614

Adjacent sequences: A320474 A320475 A320476 * A320478 A320479 A320480

Keyword

sign

Author

N. J. A. Sloane, Oct 14 2018

Status

approved