A New Version of Distributional Chaos, Distributional Chaos in a Sequence, and Other Concepts of Chaos

Abstract

In this paper, we investigate the relations between distributional chaos in a sequence and distributional chaos ([Formula: see text]-chaos, R–T chaos, DC3, respectively). Firstly, we prove a sufficient condition that the distributional chaos is equivalent to the distributional chaos in a sequence. Besides, we prove that distributional chaos in a sequence and [Formula: see text]-chaos (R–T chaos, DC3, respectively) do not imply each other. Finally, we give a new definition of chaos, named DC2[Formula: see text], which is similar to DC2, and show that DC2[Formula: see text] is an invariant of topological conjugacy and an iteration invariant (that is, for any integer [Formula: see text], [Formula: see text] is DC2[Formula: see text] if and only if [Formula: see text] is DC2[Formula: see text]).

Synchronization of Boolean networks with chaos‐driving and its application in image cryptosystem

“…For chaotic systems used in the synchronization of secure communication, the system dimension is always high and inconstant. To overcome this drawback, the one-way spatiotemporal coupled map lattice, as depicted in Equation (1), is utilized in this paper.…”

“…where G = 𝜀F (z n (N ))+(𝜀 − 1)𝜇z 2 n (1) is the defined transmission signal and G ′ is the corresponding received signal. Because the signal G is transmitted from the iteration equation of variable z n (1) and signal G ′ is received in the iteration equation of variable s n (1), the following assumption should be given before synchronization.…”

“…The master-slave coupled map lattices (2) and (3) will be synchronized with the transmission signal G and limited noise, when the coupling strength 1 > 𝜀 > 0.75, where 𝜇 ∈ (3.569, 4(1 − o)). Proof 1. : By defining the errors e n (i ) = s n (i ) − z n (i ), i = 1, 2, … , N , one can get the error e n+1 (1) = ((1 − 𝜀)𝜇+o)e n (1). Because of the limited error, the noise factor o is tiny and closed to zero.…”

“…In order to protect the privacy and ensure the security of image transmission, the synchronization results are utilized in the image cryptosystem to achieve compression and encryption. In the last few years, because of multiple complexities like initial value sensitivity, pseudo‐randomness and ergodicity [1–3], chaos has been a hot research field and has been applied in many aspects [4–6]. Especially, chaos signal can be utilized to mask the image information and is hard to identify in transmission, so it is widely applied in image encryption and transmission [7–9].…”

“…In the last few years, because of multiple complexities like initial value sensitivity, pseudo-randomness and ergodicity [1][2][3], chaos has been a hot research field and has been applied in many aspects [4][5][6]. Especially, chaos signal can be utilized to mask the image information and is hard to identify in transmission, so it is widely applied in image encryption and transmission [7][8][9].…”