preview

Nt1310 Unit 2 Math Paper

Decent Essays
Open Document

Another way to visualize the pseudo-randomness properties of the $k$-logistic map is by means of the phase diagram, or Poincar\'{e} map, which maps the value of $x^k_{t+1}$ against $x^k_{t}$ of a given orbit, or even three-dimensional by mapping $x^k_{t+2}$, $x^k_{t+1}$ against $x^k_t$. Thus, Fig.~\ref{fig:poincareLogistic} shows the phase diagrams of the $k$-logistic map for three different orbits $\mathcal{O}^{k}$. We can observe that the phase space is filled, i.e. it almost produces all possible values, while $k$ increases. \begin{figure}[h!] \centering {\includegraphics[width=8.2cm,height=14cm]{img/PoincareMapaLogisticovertical.pdf}} \caption{Phase diagrams for the $k_0$, $k_1$, $k_2$ and $k_{20}$-logistic map using $\mu = 4$. Two- and …show more content…

We assume that the $k$-logistic map gets effectively ergodic. \subsection{Number of iterations to reach chaos regime} %its perturbed part We calculated the number of iterations $\tau$ needed to achieve $|f^{\tau}(x) - f^{\tau}({x^\prime})| > 10^{-1}$ when given two initial conditions $x_0$ and $x^{\prime} =x_0 + \epsilon$ separated apart $\epsilon = 10^{-d}$. Here, the exponent $d$ represents the number of digits of precision $d = \lfloor \log |x_0 - x^{\prime}| \rfloor$. Fig.~\ref{fig:precisiondelta} depicts a log-log plot of data obtained from several average random initial conditions and its initial distance $\epsilon$ for different digits precision $d=\{10, 20, \ldots, 290\}$. Based on this plot, we found the linear approximation $\tau \sim 3.311 d$, which suggests that the chaos regime can be achieved almost three times slower when nearby close trajectories with $d$ digits of precision. Therefore, recalling to the variable $k$-right digits of the logistic map, this variation would

Get Access
Related
Get Access