Absolutely Unpredictable Chaotic Sequences

Abstract

We study chaotic functions that are exact solutions to nonlinear maps. A generalization of these functions cannot be expressed as a map of type X n+1 = g(X n , X n−1 , . . . , X n−r+1 ).The generalized functions can produce truly random sequences. Even if the initial conditions are known exactly, the next values are in principle unpredictable from the previous values. Although the generating law for these random sequences exists, this law cannot be learned from observations.

Autonomous choices among deterministic evolution–laws as source of uncertainty

“…It suggests that the Lyapunov exponent λ β is a continuous function of β over the rationals, and indeed we can prove [25] that for every β > 1, λ β = log β for almost all x 0 ∈ [0, 1]. This interpolates the obtained Lyapunov exponents for those cases when β ∈ N >1 [20]. However, this quantity is not a faithful estimate of the complexity of the system for β / ∈ N >1 , since it does not give information on the sequence of maps used along the orbit.…”

“…for some finite positive integer m (note that in Refs. [20] and [21] it was considered that g n = g ∀n, which is too restrictive to address this issue [22]). This interpretation should imply that we could produce some kind of "truly random sequences", for example, by programming an algorithm for Eq.…”

A Summary of image Encryption Algorithm Based on Chaotic Sequence

“…The cat mapping [5] was first introduced by Arnold. It is so named because it is often demonstrated using a cat face.…”

Intrinsic chaos and external noise in population dynamics

“…In Section 3 we recall the definition and properties of the random sequences given in Ref. [20,21,22,23], and we compute the complexity for the random sequences and a particular random map. In Section 4 we discuss a nonlinear time-series model constructed from sheep population data and we compute the complexity for this model.…”