A new 2D Hénon-logistic map for producing hyperchaotic behavior

Derived from the two-dimensional (2D) Hénon map and the one-dimensional (1D) Logistic map, this paper proposes a new 2D hyperchaotic map, called the 2D Hénon-Logistic map (2D-HLM). The dynamics of the 2D-HLM are investigated by means of equilibria, stability analysis, trajectory, Lyapunov exponent, and bifurcation diagram. Mathematical analysis reveals that the 2D-HLM has four unstable equilibria. Besides that, it has wide chaotic and hyperchaotic behaviors with very limited periodic windows. To evaluate the complexity performance of the 2D-HLM, Approximate entropy is used to analyze its time series. Consequently, the 2D-HLM exhibits extremely complex nonlinear behavior. With all of these attributes, the 2D-HLM would be very appropriate to produce a pseudo-random number generator that can be used in chaos-based cryptographic applications

Complexity and dynamic characteristics of a new discrete-time hyperchaotic model

Based on two of the existing one-dimensional chaotic maps and the two-dimensional Hénon map, a new two-dimensional Hénon-Gaussian-Sine model (2D-HGSM) is proposed. Basic dynamic characteristics of the 2D-HGSM are studied from the following three aspects: trajectory, bifurcation diagram and Lyapunov exponents. The complexity of 2D-HGSM is investigated by means of Approximate entropy. Performance evaluations show that the 2D-HGSM has higher complexity level, better ergodicity, wider chaotic and hyperchaotic region than different chaotic maps. Furthermore, the 2D-HGSM exhibits a qualitatively different chaotic behavior with respect to the variation of its corresponding parameters. Therefore, the 2D-HGSM has good application prospects in secure communication

Degenerating the butterfly attractor in a plasma perturbation model using nonlinear controllers

In this work, the dynamical behaviors of a low-dimensional model, which governs the interplay between a driver associated with pressure gradient and relaxation of instability due to magnetic field perturba- tions, are investigated. Besides that, two nonlinear controllers are constructed precisely to shift the equi- libria of the plasma model apart from each other. Simulation results show that shifting the equilibria can change the spacing of chaotic attractors, and subsequently break the butterfly wings into one or two symmetric pair of coexisting chaotic attractors. Furthermore, stretching the equilibria of the system apart enough from each other gives rise to degenerate the butterfly wings into several periodic orbits. In ad- dition, with appropriate initial conditions, the complex multistability behaviors including the coexistence of butterfly chaotic attractor with two point attractors, the coexistence of transient transition chaos with completely quasi-periodic behavior, and the coexistence of symmetric Hopf bifurcations are also observed

Designing new chaotic and hyperchaotic systems for chaos-based cryptography

The core of chaos-based cryptography is the selection of a good chaotic system. Most of chaotic ciphers have neglected the investigation of existence multistability in the employed chaotic systems. Meanwhile, many chaotic ciphers have applied chaotic systems with complex mathematical structure and limited chaotic behavior. Therefore, this thesis focuses on designing new chaotic and hyperchaotic systems with simple mathematical structure and complex dynamics, and discusses their performance in cryptographic applications. Furthermore, this thesis investigates the effect of existence multistability in the proposed systems from a cryptographic point of view. In the beginning, this thesis presents a new 2D discrete hyperchaotic system. Dynamic characteristics of the 2D system are investigated from the following aspects: stability, trajectory, bifurcation diagram, Lyapunov exponents and sensitivity dependence. Moreover, the complexity performance of the system is evaluated by Sample Entropy algorithm. Simulation results show that the new system has a wide hyperchaotic range with high complexity and sensitivity dependence. To investigate its performance in terms of security, a new chaos-based image encryption algorithm is also proposed. In this algorithm, the essential requirements of confusion and diffusion are accomplished, and a stochastic sequence is used to enhance the security of encrypted image. Security analysis shows that the new algorithm has good security performance.This thesis further proposed an M-dimension model as a methodological framework for producing new high-dimensional discrete hyperchaotic systems. Mathematical analysis demonstrates that the generated systems by this model have either no equilibria, or an arbitrarily large number of unstable equilibria. Moreover, numerical results show that the generated systems for certain values of parameters can produce two different behaviors: 1) single hyperchaotic attractor with high complexity and sensitivity dependence; and 2) coexistence of four attractors including single limit cycle, cluster of limit cycles, single hyperchaotic attractor, and cluster of hyperchaotic attractors, which is unusual behavior in discrete systems. However, we propose a simple feedback controller to change the chaos degradation in the multistability region from limit cycle to hyperchaotic behavior. Additionally, this thesis presents a new 4D continuous chaotic system, which is derived from Lorenz-Haken equations. Dynamics analysis, including stability of symmetric equilibria and the existence of multiple Hopf bifurcations on these equilibria, are investigated, and the coexistence of two and three different attractors is numerically revealed. Moreover, a conducted research on the complexity of the new system reveals that the complexity of a system time series can locate the parameters and initial conditions that exhibit multistability behaviors. Besides that, randomness test results demonstrate that the generated pseudo-random sequences from the multistability regions fail to pass most of the statistical tests. Finally, to choose valid pseudo-random sequences from multistability regions, this thesis constructs a new algorithm based on a new 3D multi-attribute chaotic system exhibiting extreme multistability behaviors. Unlike the existing algorithms, the proposed algorithm keeps the parameters constant with varying the initial conditions that show no non-chaotic behaviors. That means, the generated sequences are either chaotic or coexistence of chaotic attractors. Randomness test results show that the generated pseudo-random sequences by the new algorithm can pass all the statistical tests

Designing an M-dimensional nonlinear model for producing hyperchaos

This paper proposes an M-dimensional nonlinear hyperchaotic model (M-NHM) for producing new discrete-time systems with complex hyperchaotic behaviors. The M-NHM is constructed by designing an M-dimensional nonlinear system (M ≥ 2) to generate a chaotic behavior. To enhance the nonlinearity of M-NHM, hence changing its behavior to hyperchaotic, an iterative chaotic map with infinite collapse (ICMIC) is composed. Mathematical analysis shows that the M-NHM has either no equilibria, or an arbitrarily large number of equilibria. Moreover, Routh−Hurwitz criterion reveals that all these equilibria are unstable when M ≥ 3. To investigate the dynamical properties and complexity of the M-NHM, we provide 2-NHM and 3-NHM as typical examples. Simulation results show that the 2-NHM and 3-NHM have good ergodicity, wide hyperchaotic behavior, highly sensitivity dependence, and high complexity. With all these features, the M-NHM would be an ideal model for secure communications and other engineering applications

Enhancing the sensitivity of a chaos sensor for Internet of Things

Implementing chaotic systems in various applications such as sensors and cryptography shows that the sensitivity and complexity of these systems are highly required. Beside that, many existing chaotic systems exhibit low sensitivity, limited chaotic or hyperchaotic behavior, and low complexity, and this can give a negative effect on the chaos-based sensors applications. To address this problems, we present a cosine chaotification technique to enhance the chaotic characteristics of discrete systems. The proposed technique applies the cosine function as nonlinear transform to the output of a discrete system. As a typical example, we apply it on the classical 2D Hénon map. Performance evaluations show that the proposed technique can change the chaotic and non-chaotic states of the 2D Hénon map to the hyperchaotic state with extremely high complexity performance. Additionally, sensitivity dependence results, such as cross-correlation coefficient, number of non-divergent trajectories, and the change of complexity demonstrate that the enhanced Hénon map has higher sensitivity than the classical map. That means, the proposed technique would be very useful to enhance the employed systems in chaos-based sensors applications

Image Encryption Based on Local Fractional Derivative Complex Logistic Map

Local fractional calculus (fractal calculus) plays a crucial role in applications, especially in computer sciences and engineering. One of these applications appears in the theory of chaos. Therefore, this paper studies the dynamics of a fractal complex logistic map and then employs this map to generate chaotic sequences for a new symmetric image encryption algorithm. Firstly, we derive the fractional complex logistic map and investigate its dynamics by determining its equilibria, geometric properties, and chaotic behavior. Secondly, the fractional chaotic sequences of the proposed map are employed to scramble and alter image pixels to increase resistance to decryption attacks. The output findings indicate that the proposed algorithm based on fractional complex logistic maps could effectively encrypt various kinds of images. Furthermore, it has better security performance than several existing algorithms