The asymptotic deterministic randomness

Statistical independence in nonlinear maps coupled to non-invertible transformations

“…(1) presents fundamental differences with the previously known chaotic systems. This phenomenon is called "deterministic randomness" [18,[24][25][26][27][28]. A very interesting development in the study of "deterministic randomness" has been the construction of the so-called Lissajous maps [24][25][26][27][28].…”

“…Given some conditions for z and b, the Lissajous maps have been shown to produce exactly the return maps that are generated by the random processes defined by Eq. (1) [24][25][26][27][28]. Consider the following particular example:…”

“…The Lissajous maps have been utilized in important applications, including random number generation and cryptography [24][25][26][27][28]. However, in most cases, the used function f (•) has been trigonometric, e.g., f (•) = cos(•), f (•) = sin 2 (•), etc.…”

“…Recently the theory of functions and dynamical systems that can produce unpredictable time-series has been a very active area of research [15][16][17][18][19][20][21][22][23][24][25][26][27][28]. Some of these works are based on ideas as the following.…”

“…(1) are the solutions to several classes of chaotic maps [15][16][17][18][19][20][21][29][30][31][32][33][34][35]. Several dynamical systems have been proposed to generate similar dynamics [17][18][19][20][21][22][23][24][25][26][27][28]. In the present Letter we will investigate the higher-order correlations of nonlinear maps coupled to noninvertible transformations.…”