Research on 4-dimensional Systems without Equilibria with Application

Abstract

Recently chaos-based encryption has been obtained more and more attention. Chaotic systems without equilibria may be suitable to be used to design pseudorandom number generators (PRNGs) because there does not exist corresponding chaos criterion theorem on such systems. This paper proposes two propositions on 4-dimensional systems without equilibria. Using one of the propositions introduces a chaotic system without equilibria. Using this system and the generalized chaos synchronization (GCS) theorem constructs an 8-dimensional discrete generalized chaos synchronization (8DBDGCS) system. Using the 8DBDGCS system designs a 216-word chaotic PRNG. Simulation results show that there are no significant correlations between the key stream and the perturbed key streams generated via the 216-word chaotic PRNG. The key space of the chaotic PRNG is larger than 21275. As an application, the chaotic PRNG is used with an avalanche-encryption scheme to encrypt an RGB image. The results demonstrate that the chaotic PRNG is able to generate the avalanche effects which are similar to those generated via ideal chaotic PRNGs.

Chaos synchronization in a 6-D hyperchaotic system with self-excited attractor

“…Therefore, this control is failed. If update the matrix with the same control as: (14) where 2 = (10,1,8/3,1,5/42,1) is a positive definite. Figure 3 shows verify these results numerically.…”

“…In 1979, Rössler discovers the first 4-D hyperchaotic system including real variables with two positive Lyapunov exponents and followed to discover another 4-D, as well as 5-D hyperchaotic with three positive Lyapunov exponents [10,[13][14][15] and some other systems, have been revealed. The dynamical systems with higher dimensions are effective and interesting compared with the low dimensions [16][17][18].…”

Projective and hybrid projective synchronization of 4-D hyperchaotic system via nonlinear controller strategy

“…In nonlinear dynamic systems, chaos synchronization is the first phenomenon which discovered by Fujisaka and Yamada in 1983, but did not receive great interest until 1990 when Pecora and Carrol developed this phenomenon between two identical chaotic systems with different initial condition [1][2][3][4]. Chaos synchronization has attracted considerable attention due to its important applications in physical systems [1], biological systems [5], Encryption [6] and secure communications [7], etc. After then, several attempts were made to create many types of synchronization phenomena such as Complete Synchronization (CS) [2,4,8], Anti-Synchronization (AS) [9,10], Hybrid Synchronization (HS) [11], Generalized Synchronization (GS) [12], Projective Synchronization (PS) [13], Hybrid Projective Synchronization (HPS) [14] and Generalized Projective Synchronization (GPS) [15].…”

Projective synchronization for a cass of 6-D hyperchaotic lorenz system

“…The projective synchronization and generalized projective synchronization are based on nonzero constant (scaling factor) and constant scaling matrix respectively, therefore, projective synchronization includes three strategies: full synchronization, anti-synchronization and hybrid synchronization are special cases where the scaling factor = , = and = respectively [6][7][8][9]. Chaotic system has become an important subject in study of behaviors of dynamical system,s [10][11][12][13]. But this system has contains one positive Lyapunov exponent only while hyperchaotic system has more than one positive Lyapunov exponent.…”

“…So, it's needed to propose high dimensional nonlinear dynamical systems, these system are characterized as a chaotic system with more than one positive Lypunov exponent, and have more complex and richer dynamical behaviors than chaotic system. Historical, Rössler system is the first hyperchaotic systems which discover in 1979, Since then, many hyperchaotic systems have been discover [17][18][19][20][21][22] [23,24]. There are many works deals with synchronization phenomena for various different dimension nonlinear dynamical systems (3-D, 4-D and 5-D) and a few researchers consider system with six dimension.…”