A320475 - OEIS

%I #16 Oct 22 2018 14:36:25

%S 1,4,5,5,4,1,0,-1,-5,-6,-6,-5,-3,-1,0,-1,-2,-5,-6,-7,-9,-12,-14,-18,

%T -19,-23,-25,-25,-27,-31,-33,-36,-38,-41,-43,-44,-46,-49,-51,-55,-57,

%U -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

%N 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.

%C 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.

%C Note that some websites reduce x(n) mod 2*Pi, but this version does not.

%D H. A. Lauwerier, Two-dimensional iterative maps, Chapter 4 of A. V. Holden, ed., Chaos, Princeton, 1986. See Eq. (4.67).

%D E. N. Lorenz, The Essence of Chaos, Univ. Washington Press, 1993. See p 191.

%H Roderick V. Jensen, <a href="https://www.jstor.org/stable/27854538">Classical chaos</a>, American Scientist 75.2 (1987): 168-181. See Eq. (2), (3).

%e The initial values of x(n) and y(n) are

%e 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, ...

%e and

%e 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, ...

%p k:=2; M:=120; x[0]:=1; y[0]:=1;

%p for n from 1 to M do

%p y[n]:=y[n-1]+k*evalf(sin(x[n-1]));

%p x[n]:=x[n-1]+y[n];

%p od:

%p [seq(x[n],n=0..M)];

%p [seq(y[n],n=0..M)];

%p [seq(round(x[n]),n=0..M)]; # A320475

%Y Cf. A320472-A320480.

%K sign

%O 0,2

%A _N. J. A. Sloane_, Oct 14 2018