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How to create chaotic sequence by lorenz maps and fourth-order Runge– Kutta method?

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Arshub
Arshub on 28 Dec 2021
Closed: John D'Errico on 28 Dec 2021
How to create chaotic sequence from equation bellow by fourth-order Runge– Kutta method and what does "S (i) "mean?
suppos x0,y0,z0,w0 any initial values.
Lorenz system is a 3-D continuous chaotic system, and it is given by Eq. (1):
x˙ = a (y − x)
y˙ = cx − y − xz
z˙ = xy − bz
where a, b and c are parameters of the chaotic system, system is going to be chaotic. By adding a nonlinear when parameters a = 10, b = 8/3, and c = 28, the Lorenz system is going to be chaotic. By adding a controller ˙w =−yz + γw to Eq. (1), hyper-chaotic Lorenz system is given by:
x˙ = a (y − x) +w
y˙ = cx − y − xz
z˙ = xy − bz
w˙ =−yz + γw
where a, b, c and γ are parameters of hyper-chaotic system. When system parameters a = 10, b = 8/3, c = 28 andγ ∈ [−1.52,−0.06],
Under the initial conditions x0, y0, z0 and w0,Eq. (2) iterated by using fourth-order Runge– Kutta method with 0.002 step size, and the iteration of x is denoted asX. The key stream S generated by iteration sequence X as:
S (i) = mod( fix (X (i) + 100) *(10^16),256)
where mod (a, b) means the remainder of a/b,fix(a) takes the integer part of a, i is the serial number of elements in the sequences S and X.

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