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In the standard logistic map $x_{n+1} = r\,x_n (1-x_n)$, for at least most $r$ in range $[3.57,4)$ and $x_0\in(0,1]$, the behavior is chaotic (that is, a small change in $x_0$ amplifies as we iterate). My restriction of $x_0$ to $[1/4,3/4)$ is chiefly to avoid $x_0=0$, which is stationary, but $x_0=(k+1)/2^{128}$ would also work. I can't tell for the 2D hyper-chaotic system used in the article. At least, using two variables vastly increases the state space, which is dangerously small with a single IEEE-754 64-bit float.

In the standard logistic map $x_{n+1} = r\,x_n (1-x_n)$, for at least most $r$ in range $[3.57,4)$ and $x_0\in(0,1]$, the behavior is chaotic (that is, a small change in $x_0$ amplifies as we iterate). My restriction of $x_0$ to $[1/4,3/4)$ is chiefly to avoid $x_0=0$ and $x_0=1$, which get stuck to $0$. I can't tell for the 2D hyper-chaotic system used in the article. At least, using two variables vastly increases the state space, which is dangerously small with a single IEEE-754 64-bit float.

As of $10^{15}$, my guess is the authors used the power of ten below $2^{52}$, where $52$ is the number of bits in the mantissa in an IEEE-754 64-bit floating point number†. This errs comfortably on the safe side, and happens to more than compensate the error made by simply multiplying $0.43$ and $10^15$, ignoring that for a value $\mu\in[3.57,4)$, the two high-order bits of the mantissa always are set. There are actually about $2^{52}\times0.43/(4-2)\approx0.968\times10^{15}$ distinct IEEE-754 64-bit float in $[3.57,4)$.

In the standard logistic map $x_{n+1} = r\,x_n (1-x_n)$, for at least most $r$ in range $[3.57,4)$ and $x_0\in(0,1]$, the behavior is chaotic (that is, a small change in $x_0$ amplifies as we iterate). My restriction of $x_0$ to $[1/4,3/4)$ is chiefly to avoid $x_0=0$, which is stationary, but $x_0=(1+k)/2^{128}$ would also work. I can't tell for the 2D hyper-chaotic system used in the article. At least, using two variables vastly increases the state space, which is dangerously small with a single IEEE-754 64-bit float.

As of $10^{15}$, my guess is the authors used the power of ten below $2^{52}$, where $52$ is the number of bits in the mantissa in an IEEE-754 64-bit floating point number†. This errs comfortably on the safe side, and happens to more than compensate the error made by simply multiplying $0.43$ and $10^{15}$, ignoring that for a value $\mu\in[3.57,4)$, the two high-order bits of the mantissa always are set. There are actually about $2^{52}\times0.43/(4-2)\approx0.968\times10^{15}$ distinct IEEE-754 64-bit float in $[3.57,4)$.

In the standard logistic map $x_{n+1} = r\,x_n (1-x_n)$, for at least most $r$ in range $[3.57,4)$ and $x_0\in(0,1]$, the behavior is chaotic (that is, a small change in $x_0$ amplifies as we iterate). My restriction of $x_0$ to $[1/4,3/4)$ is chiefly to avoid $x_0=0$, which is stationary, but $x_0=(k+1)/2^{128}$ would also work. I can't tell for the 2D hyper-chaotic system used in the article. At least, using two variables vastly increases the state space, which is dangerously small with a single IEEE-754 64-bit float.

Update:

$x_0$ and $\mu$ are used as keys for the logistic maps. They have used 64-bit double precision number. The range of $\mu$ is mentioned be $0.43 \times 10^{15}$. From where this number came?

Quoting the article

The computational precision of 64-bit double-precision number is about $10^{15}$. Therefore, the $K_p$ parameters and $x_0$ can be any value among $10^{15}$ numbers. $\mu$ can have any number from $0.43\times10^{15}$ values.

I think the $0.43$ is the width of the range of $\mu$ stated in section 3.1.2:

The system is chaotic for $\mu\in[3.57,4]$

As of $10^{15}$, my guess is the authors used the power of ten below $2^{52}$, where $52$ is the number of bits in the mantissa in an IEEE-754 64-bit floating point number†. This errs comfortably on the safe side, and happens to more than compensate the error made by simply multiplying $0.43$ and $10^15$, ignoring that for a value $\mu\in[3.57,4)$, the two high-order bits of the mantissa always are set. There are actually about $2^{52}\times0.43/(4-2)\approx0.968\times10^{15}$ distinct IEEE-754 64-bit float in $[3.57,4)$.

Why are they not restricting the space for the initial value?

In the standard logistic map $x_{n+1} = r\,x_n (1-x_n)$, for at least most $r$ in range $[3.57,4)$ and $x_0\in(0,1]$, the behavior is chaotic (that is, a small change in $x_0$ amplifies as we iterate). My restriction of $x_0$ to $[1/4,3/4)$ is chiefly to avoid $x_0=0$, which is stationary, but $x_0=(1+k)/2^{128}$ would also work. I can't tell for the 2D hyper-chaotic system used in the article. At least, using two variables vastly increases the state space, which is dangerously small with a single IEEE-754 64-bit float.

† which I linked to before the article was stated, because it's so typical of work using the logistic map for encryption. This one is representative of a sub-genre of the cryptographic literature: encryption specialized to digital image for no discernible reason, with visual illustrations as security argument (often complemented by some statistical indicator). There's no sufficiently precise description to exactly replicate the system, which is a strong guarantee that it won't be attacked. Many similar articles are cited, and the article itself gets highly cited, which seems to be part of the game played.