alain@elevia.UUCP (W.A.Simon) (06/22/91)
In <1991Jun20.194552.15875@cunews.carleton.ca> rdb@scs.carleton.ca (Robert D. Black) writes: > I recently read that the chaotic logistic equation > u(t+1) = 4u(t)(1-u(t)) u(0) in 0..1, > has an ANALYTIC SOLUTION: > u(t) = sin**2 (2**(t-1) arccos(1-2u(0))) > This is CONFUSING! Wasn't it the case that solvable systems > are by definition predictable and hence not chaotic? Here you > can find the value of the system at any time t without computing > intermediate values. Yet the logistic equation above is said to be > chaotic! Would it be that your equation has just been proven to be non chaotic, or would it be that chaos-order is a continuum, and that Laplace was right? What we perceive as chaos is just a weakness in our instrumentation... or computing power. Half of a |8-) My interest in chaotic sequences is due to the belief that Poincare could be right, and therefore I could use such a sequence to generate cryptanalytically strong keys. Half of a |8-( Which brings me back to the Ulam sequences we discussed the other day (aka hailstone numbers). Are the odd/even transitions in the sequences known to contain identifiable patterns? In other words, would a string of 0's and 1's matching, respectively, even numbers and odd numbers in the sequence, be considered to be a random bit stream? > Robert Black -- William "Alain" Simon UUCP: alain@elevia.UUCP
alain@elevia.UUCP (W.A.Simon) (06/22/91)
In <1991Jun22.133638.3258@elevia.UUCP> alain@elevia.UUCP (W.A.Simon) writes: > Which brings me back to the Ulam sequences we discussed the > other day (aka hailstone numbers). Are the odd/even transitions > in the sequences known to contain identifiable patterns? In > other words, would a string of 0's and 1's matching, respectively, > even numbers and odd numbers in the sequence, be considered to be > a random bit stream? Before I get skewered on a Hilbert curve, let me rephrase this. Would the tools of statistical analysis (Chi-Square, etc...) identify that this sequence is not random ? -- William "Alain" Simon UUCP: alain@elevia.UUCP
hrubin@pop.stat.purdue.edu (Herman Rubin) (06/23/91)
In article <1991Jun22.140127.3984@elevia.UUCP>, alain@elevia.UUCP (W.A.Simon) writes: > In <1991Jun22.133638.3258@elevia.UUCP> alain@elevia.UUCP (W.A.Simon) writes: .................... > Before I get skewered on a Hilbert curve, let me rephrase > this. Would the tools of statistical analysis (Chi-Square, > etc...) identify that this sequence is not random ? If you ask whether the marginal distribution approaches the limiting one, Beta(.5,.5) for the particular example, the answer is yes. If you actually tested it using a two-sided test, you would even find the sample distribution converged too fast, but it would take quite a large sample to detect that. But if you looked at pairs, they all lie on a very simple curve, which is very obvious. The chaotic nature is that a slight difference at one point makes a big difference a considerable time in the future. -- Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 Phone: (317)494-6054 hrubin@l.cc.purdue.edu (Internet, bitnet) {purdue,pur-ee}!l.cc!hrubin(UUCP)