Nonlinear Sciences – Chaotic Dynamics
Scientific paper
1998-03-10
Nonlinear Sciences
Chaotic Dynamics
Replaced to conform with version accepted by Complexity
Scientific paper
A number of observations are made on Hofstadter's integer sequence defined by Q(n)= Q(n-Q(n-1))+Q(n-Q(n-2)), for n > 2, and Q(1)=Q(2)=1. On short scales the sequence looks chaotic. It turns out, however, that the Q(n) can be grouped into a sequence of generations. The k-th generation has 2**k members which have ``parents'' mostly in generation k-1, and a few from generation k-2. In this sense the series becomes Fibonacci type on a logarithmic scale. The mean square size of S(n)=Q(n)-n/2, averaged over generations is like 2**(alpha*k), with exponent alpha = 0.88(1). The probability distribution p^*(x) of x = R(n)= S(n)/n**alpha, n >> 1, is well defined and is strongly non-Gaussian. The probability distribution of x_m = R(n)-R(n-m) is given by p_m(x_m)= lambda_m * p^*(x_m/lambda_m). It is conjectured that lambda_m goes to sqrt(2) for large m.
No associations
LandOfFree
Order and Chaos in Hofstadter's Q(n) Sequence does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Order and Chaos in Hofstadter's Q(n) Sequence, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Order and Chaos in Hofstadter's Q(n) Sequence will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-403483