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Based on the paper titled, Simultaneous Arithmetic Coding and Encryption Using Chaotic Maps

and another related paper http://www-connexe.univ-brest.fr/lest/tst/publications/pdf/comm04_compression_chaos.pdf

Authors use a piecewise linear chaotic map for their application which is encoding a bitstream into the chaotic trajectory of the map. In the second paper, they say that the map is a Bernoulli map. I am not familiar with piecewise linear chaotic Map. Is it an extension of the Skew Tent Map or the Tent map? The Tent Map is $$x_{n+1} = a(1-|2 x_n -1|)$$ where $a \in [0,1]$ and $x_n \in [0,1]$

Are the Bernoulli map, Tent Map and the piecewise linear map the same in terms of the derivatives of the map and properties? Can I say that the piecewise Tent Map is obtained by extending the Skew map or the Tent map?

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These are three different maps, if you call

$$ x_{n+1} = f(x_n) $$

then the function $f$ is different for all the naps you mentioned, and so is the dynamics they generate. More precisely, the skewed tent map can be reduced to the tent map by a particular choice of parameters

  • Tent map $$ f(x) = a(1 - |2x - 1|) $$

  • Bernoulli shift map

    $$ f(x) = \begin{cases} 2x, & \mbox{for}\quad 0\le x \le 1/2 \\ 2x - 1, & \mbox{for}\quad 1/2< x \le 1\end{cases} $$

  • Skew tent map

    $$ f(x) = \begin{cases} \nu + (1 - \nu)x/\mu, & \mbox{for}\quad 0\le x \le \mu \\ (1 - x) / (1- \mu), & \mbox{for}\quad \mu< x \le 1\end{cases} $$

    Note that the tent map can be recovered by setting $\nu =0$ and $\mu = 1/2$

The figure below shows a plot of $f(x)$ for these three cases

enter image description here

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  • $\begingroup$ Thank you for the illustration as well. Could you please also help in explaining these points (A) Can I consider the piecewise linear chaotic map in the paper to be the extension of the Skew Tent Map by saying that it is derived from higher order iterates of Tent map? (B) If the partial derivative (jacobian ) of the Tent map is $F_n = 2a$ for $x_n < 0.5$ and $-2a$ for $x_n > 0.5$ then could you please let me know what would it be for the Skew Tent Map and would it be the same for the piecewise linear chaotic map? $\endgroup$ – SKM Nov 26 '16 at 19:15
  • $\begingroup$ @SKM I fail to see a smooth transformation between these two maps. To calculate the Jacobbian just take the derivative with respect to $x$ in the last equation I showed $\endgroup$ – caverac Nov 26 '16 at 19:23
  • $\begingroup$ I did find that but I am having difficulty in mentioning the range as is available for the Tent map that I gave in the previous comment. Assuming $v=0$, the partial order first derivative is $F(x) = 1/\mu$ and $F(x) = 1$ is there a condition for $x$ to be less than something for the derivative to exist? $\endgroup$ – SKM Nov 26 '16 at 19:34
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    $\begingroup$ $F(x) = 1/\mu$ for $0<x<\mu$ and $F(x) = 1 / (\mu - 1)$ for $\mu<x<1$ $\endgroup$ – caverac Nov 26 '16 at 19:38

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