Chaos, Periodic Chaos, and the Random- Walk Problem

The authors have studied whether numerically generated sequences from the logistic parabola $f_b\mathrm{}\left(x\right)=4\mathrm{bx}(1-x)\mathrm{}$ with $b$, $x\in \mathrm{}[0,1]\mathrm{}$, for values of $b$, above the Feigenbaum critical value $b_{\infty }$, are truly chaotic or whether they are periodic but with exceedingly large periods and very long transients. Using the logistic parabola the authors calculate via Monte Carlo simulation the average walk length for trapping on a one-dimensional lattice with a centrosymmetric trap. Comparison with exact results suggests that the only "truly chaotic" sequence is the one for which $b=1\mathrm{}$.