The number of iterations for information loss in the 3x mod 1 chaotic map

2018-10-13 21:48:50

We want to deduce after how many iterations we can predict the state of some system (in this case, 3x mod 1). We are given that we know the initial state of some f(x) to within 10^-7 and want to know after how many iterations of 3x mod 1 we can still predict the state to within 10^-3 accuracy.

We are also given that after each iteration, we lose exactly 1 bit of information.

My assumption is that the approach to solving this problem involves the binary expansion of numbers. Each such number has form x = a1a2a3..... where each ai represents the 2^-i contribution to x. Then after each iteration we lose exactly one decimal place, one 2^-i contribution. But then if we know the initial state to 10^-7, to how many decimal places do we know the binary expansion of x? If we knew it to 2^-7, that would be straightforward, but I'm not actually sure how this translates.

Furthermore, this binary expansion is essentially the application of the 2x mod 1 map, counting 1 bit when the

  • One way to approach it is to ask if we measure the starting location within $\pm 10^{-7}$ how many iterations does it take for the error to perhaps exceed $\pm 10^{-3}$. As we triple at each iteration, after $n$ iterations the error could be as large as $\pm 3^n \cdot 10^{-7}$ so we want to solve $$3^n \gt 10^4\\n \gt \log_3 (10^4)\approx 8.38\\n \ge 9$$

    so it takes $9$ iterations for the error to exceed $10^{-3}$

    The iteration $3x \bmod 1$ loses one base $3$ digit every iteration, not one bit.

    2018-10-13 23:25:02