Since its original publication 1 in 1978, Lozi’s chaotic map has been thoroughly explored and continues to be. Hundreds of publications analyze its particular structure or apply its properties in many fields (electronic devices like memristor, A.I. with swarm intelligence, etc.). Several generalizations have been proposed, transforming the initial two-dimensional map into multidimensional one. However, they do not respect the original constraint that allows this map to be one of the few strictly hyperbolic: a constant Jacobian. In this paper we introduce a three-dimensional piece-wise linear extension respecting this constraint and we explore a special property never highlighted for chaotic mappings: the coexistence of thread-chaotic attractors (i.e., attractors which are formed by collection of lines) and sheet-chaotic attractors (i.e., attractors which are formed by collection of planes). This new 3-dimensional mapping can generate a large variety of chaotic and hyperchaotic attractors. We give five examples of such behavior in this article. In the first three examples, there is coexistence of thread and sheet-chaotic attractors. However, their shape are different and they are constituted by a different number of pieces. In the two last examples, the blow up of the attractors with respect to parameter a and b is highlighted.

With the explosion of the generation, transmission and sharing of image data over the Internet and other unsecured networks, the need and significance for the development of novel image encryption algorithms is unprecedented. In this research work, we propose a novel framework for image encryption that is based on two hyperchaotic maps, utilized in conjunction with the single neuron model (SNM). The framework entails three successive stages, where in every stage, a substitution box (S-box) is applied, then XORing with an encryption key is carried out. The S-boxes and the encryption keys are generated from the numerical solutions of the hyperchaotic maps and the SNM. The performance of the proposed framework is gauged through a number of metrics, reflecting superior performance and complete asymmetry between the plain images and their encrypted versions. However, the main advantages of this work are: (1) The attained width of the key space; and (2) The achieved efficiency in software implementation. The superior key space of 2^{2551} is the result of employing the two hyperchaotic maps, while the improved efficiency, resulting in an average encryption rate of 8.54 Mbps, is the result of the utilization of the single neuron model, as well as the employment of optimized parallel processing techniques. In addition, the proposed encryption framework is shown to output encrypted images that pass the NIST SP 800 suite.

This paper introduces the complex Rayleigh-van der Pol-Duffing oscillators (RVDOs), which are hyperchaotic and can be either autonomous or non-autonomous. The fundamental dynamics of the autonomous and nonautonomous complex RVDOs including dissipation, symmetry, fixed points and their stability are studied. These oscillators are found in various important fields of physics and engineering. The paper proposes a scheme to achieve phase synchronization (PS) and antiphase synchronization (APS) for different dimensional models. These kinds of synchronization are considered as generalization of several other types of synchronization. For this scheme we use the active control method based on Lyapunov stability theory. By analytically determining the control functions, the scheme achieved PS and APS. Our scheme is applied to study PS of hyperchaotic behaviours for two distinct hyperchaotic non-autonomous and autonomous complex RVDOs. Additionally, the scheme is employed to achieve APS of chaotic real non-autonomous RVDO and hyperchaotic complex autonomous RVDO, including those with different dimensions. Our work presents numerical results that plot the amplitudes and phases of these hyperchaotic behaviours, demonstrating the successful achievement of PS and APS.

Chaotic theory is commonly being researched for application in image encryption scheme (IES). In this paper, a novel two-dimensional cross hyperchaotic Sine-modulation-Logistic map (2D-CHSLM) based on the famous Sine and Logistic maps. The hyperchaotic behaviors of 2D-CHSLM is testified with the help of their bifurcation diagrams, trajectory plots, Lyapunov exponents, sample entropy, C0 complexity, permutation entropy and 0-1 test. A new bit-level IES with traditional encryption structure is proposed utilizing 2D-CHSLM. The 2D-CHSLM-IES performs a 2D-CHSLM-based bit-level confusion using zig-zag transform and a 2D-CHSLM-based cross coupled diffusion operations for generating a highly secure cipher-image. Then the simulations and security analyses are further carried out with the help of key space, key sensitivity, histogram, correlation, information entropy, differential attack, noise and data loss attacks, prove that the security of the proposed 2D-CHSLM-IES

Four discrete models using the exact spectral derivative discretization finite difference (ESDDFD) method are proposed for a chaotic five-dimensional, conformable fractional derivative financial system incorporating ethics and market confidence. Since the system considered was recently studied using the conformable Euler finite difference (CEFD) method and found to be hyperchaotic, and the CEFD method was recently shown to be valid only at fractional index , the source of the hyperchaos is in question. Through numerical experiments, illustration is presented that the hyperchaos previously detected is in part an artifact of the CEFD method as it is absent from the ESDDFD models.