High-Security Image Encryption Based on a Novel Simple Fractional-Order Memristive Chaotic System with a Single Unstable Equilibrium Point

Fractional-order chaotic systems have more complex dynamics than integer-order chaotic systems. Thus, investigating fractional chaotic systems for the creation of image cryptosystems has been popular recently. In this article, a fractional-order memristor has been developed, tested, numerically analyzed, electronically realized, and digitally implemented. Consequently, a novel simple three-dimensional (3D) fractional-order memristive chaotic system with a single unstable equilibrium point is proposed based on this memristor. This fractional-order memristor is connected in parallel with a parallel capacitor and inductor for constructing the novel fractional-order memristive chaotic system. The system’s nonlinear dynamic characteristics have been studied both analytically and numerically. To demonstrate the chaos behavior in this new system, various methods such as equilibrium points, phase portraits of chaotic attractor, bifurcation diagrams, and Lyapunov exponent are investigated. Furthermore, the proposed fractional-order memristive chaotic system was implemented using a microcontroller (Arduino Due) to demonstrate its digital applicability in real-world applications. Then, in the application field of these systems, based on the chaotic behavior of the memristive model, an encryption approach is applied for grayscale original image encryption. To increase the encryption algorithm pirate anti-attack robustness, every pixel value is included in the secret key. The state variable’s initial conditions, the parameters, and the fractional-order derivative values of the memristive chaotic system are used for contracting the keyspace of that applied cryptosystem. In order to prove the security strength of the employed encryption approach, the cryptanalysis metric tests are shown in detail through histogram analysis, keyspace analysis, key sensitivity, correlation coefficients, entropy analysis, time efficiency analysis, and comparisons with the same fieldwork. Finally, images with different sizes have been encrypted and decrypted, in order to verify the capability of the employed encryption approach for encrypting different sizes of images. The common cryptanalysis metrics values are obtained as keyspace = 2648, NPCR = 0.99866, UACI = 0.49963, H(s) = 7.9993, and time efficiency = 0.3 s. The obtained numerical simulation results and the security metrics investigations demonstrate the accuracy, high-level security, and time efficiency of the used cryptosystem which exhibits high robustness against different types of pirate attacks.

Controlling Chaotic Systems via Time-delayed Control

Based on Lyapunov stabilization theory, this paper proposes a proportional plus integral time-delayed controller to stabilize unstable equilibrium points (UPOs) embedded in chaotic attractors. The criterion is successfully applied to the classic Chua's circuit. Theoretical analysis and numerical simulation show the effectiveness of this controller.

An Unstable Equilibrium

This chapter analyses the nationalist conflict in Sindh since 1988. It examines the multiple stints of the PPP in power, the dismissal of its governments at the Centre and in Sindh and the deepening of the ethnic schism between the Muhajirs and Sindhis under the Muttahida Quomi Movement (MQM). This chapter traces the maturation of Muhajir identity and its political aspirations as expressed through the agency of the MQM, from its rise in the mid-1980s to its fragmentation from 2016 onwards. The chapter analyzes the trends of ethnic violence that engulfed Sindh during this period, examines the 2018 elections in Sindh—which saw the entry into government of the Pakistan Tehreek-I-Insaaf (PTI)—and explores the 2018–20 tug-of war between Islamabad and Karachi over provincial autonomy.

Frequency-Energy Plots of Bistable Nonlinear Energy Sink

The nonlinear energy sink (NES), which is proven to perform rapid and passive targeted energy transfer (TET), has been has been employed for vibration mitigation in many primary small- and large-scale structures. Recently, the feature of bistability, in which two nontrivial stable equilibria and one trivial unstable equilibrium exist, is utilized for passive TET in what is known as Bistable NES (BNES). The BNES generates a nonlinear force that incorporates negative linear and multiple positive or negative nonlinear stiffness components. In this paper, the BNES is coupled to a linear oscillator (LO) where the dynamic behavior of the resulting LO-BNES system is studied through frequency-energy plots (FEPs), which are generated by analytical approximation using the complexification-averaging method and by numerical continuation techniques. The effect of the length and stiffness of the transverse coupling springs is found to affect the stability and topology of the branches and indicates the importance of the exact physical realization of the system. The rich nonlinear dynamical behavior of the LO-BNES system is also highlighted through the appearance of multiple symmetrical and unsymmetrical in- and out of- phase backbone branches, especially at low energy levels. The wavelet transform is imposed into the FEP for variety of initial conditions and damping content and it is found that the FEP has backbone branches at low energy levels associated with the oscillation of the bistable attachments about one of its stable equilibrium positions where passage through the unstable equilibrium position does not occur.

Impulsive control of a class of multiple unstable neural networks

This paper addresses the issue of stability of a class of multiple unstable Cohen–Grossberg neural networks(CGNNs) under impulsive control. Some novel sufficient conditions are given to make the unstable equilibrium points of the model locally μ-stable. An example is offered to demonstrate the effectiveness of the control strategy by comprehensive computer simulations.

From cash to private and public digital currencies. The risk of financial instability and “modern monetary Middle ages”

The article analyzes the unstable equilibrium between further developing national monetary systems by means of (private/public) digital currencies while maintaining financial stability as secured by “tangible” store of values like notes/coins. Which are those elements of innovation improving today’s payments system? And, at the same time, which modern trends might destabilize the above-mentioned equilibrium? The paper will identify fundamental monetary principles to be respected, no matter what the innovation level in post-modern economies might soon look like. Cryptocurrencies will be also compared to central bank digital currencies (CBDC), which might soon complement or even replace notes and coins. But, is cash truly a “barbarous relic”? And, which impact might have its legal limitation (as occurring in several European countries)?