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.

Nonintegrability of the problem of a spherical top rolling on a vibrating plane

This paper investigates the rolling motion of a spherical top with an axisymmetric mass distribution on a smooth horizontal plane performing periodic vertical oscillations. For the system under consideration, equations of motion and conservation laws are obtained. It is shown that the system admits two equilibrium points corresponding to uniform rotations of the top about the vertical symmetry axis. The equilibrium point is stable when the center of mass is located below the geometric center, and is unstable when the center of mass is located above it. The equations of motion are reduced to a system with one and a half degrees of freedom. The reduced system is represented as a small perturbation of the problem of the Lagrange top motion. Using Melnikov’s method, it is shown that the stable and unstable branches of the separatrix intersect transversally with each other. This suggests that the problem is nonintegrable. Results of computer simulation of the top dynamics near the unstable equilibrium point are presented.

A Four-Zone Model and Nonlinear Dynamic Analysis of Solution Multiplicity of Buoyancy Ventilation in Underground Building

The solution multiplicity of natural ventilation in buildings is very important to personnel safety and ventilation design. In this paper, a four-zone model of buoyancy ventilation in typical underground building is proposed. The underground structure is divided to four zones, a differential equation is established in each zone, and therefore, there are four differential equations in the underground structure. By solving and analyzing the equilibrium points and characteristic roots of the differential equations, we analyze the stability of three scenarios and obtain the criterions to determine the stability and existence of solutions for two scenarios. According to these criterions, the multiple steady states of buoyancy ventilation in any four-zone underground buildings for different stack height ratios and the strength ratios of the heat sources can be obtained. These criteria can be used to design buoyancy ventilation or natural exhaust ventilation systems in underground buildings. Compared with the two-zone model in (Liu et al. 2020), the results of the proposed four-zone model are more consistent with CFD results in (Liu et al. 2018). In addition, the results of proposed four-zone model are more specific and more detailed in the unstable equilibrium point interval. We find that the unstable equilibrium point interval is divided into two different subintervals corresponding to the saddle point of index 2 and the saddle focal equilibrium point of index 2, respectively. Finally, the phase portraits and vector field diagrams for the two scenarios are given.

Stabilization and disturbance attenuation control of the gyroscopic inverted pendulum

In this study, a control moment gyroscope is used as an actuator to stabilize the inverted pendulum. A control strategy is proposed to stabilize the inverted pendulum at the upright unstable equilibrium point and to maintain the gimbal angle as small as possible. Such a problem is formulated as a constrained H∞ disturbance attenuation problem and then transformed into solving linear matrix inequalities. The performance of the proposed controller is evaluated through simulation for linear and nonlinear cases. It is shown that the proposed state-feedback control strategy effectively stabilizes the inverted pendulum.

Derivative Analysis of Global Average Temperatures

Contemporary climate studies are quite numerous, and the topic in general has become politically charged, where opposing political interests have generated juxtaposed studies with contradictory results despite near unanimous acceptance of the fact the planet is warning.  One emotionally charged issue is whether the increase is man-made or naturally occurring.  This study avoids political pitfalls and controversial postulations, instead seeking to add to the literature a mathematical catastrophe analysis based on derivative modelling and extrapolation to deduce whether an unexpected (sudden) rise or fall in dynamic global atmospheric temperatures (e.g. an ice age) is predicted by the dynamic atmospheric temperature data.  The study concludes that extrapolations of a derivative model encounter an unstable equilibrium point at the end of this century leading to a prediction of the potential for a sudden, dramatic increase in global average temperatures. The author takes care not to make controversial predictions, instead merely follows the mathematical facts where they lead:  The potential for a catastrophe if the unnamed causes of global temperature increases remain unaddressed.

Modeling and implementation of 1-D inverted magnetic needle system using a robust sliding mode controller

In this paper, we propose a novel problem in control systems area involving the control of a magnetic needle in the presence of an external magnetic field. A magnetic needle when restricted to rotate about a single axis in an external magnetic field, by pivoting its center will produce a stable and unstable equilibrium. Here, we present the detailed mathematical modeling of the 1-D inverted magnetic needle system and its control in the unstable equilibrium point. We use sliding mode controller (SMC) to achieve the control objectives. The simulation results are validated with the experimental results. For achieving a close match, we consider sensor and actuator nonlinearities. Further, its robust performance is compared with proportional-derivative (PD), proportional-integral-derivative (PID) controllers in the presence of system parameter uncertainty, disturbance, and sensor delay. We also study the effect of change in SMC parameters, proportional and derivative gains on the system performance. It is to be noted that the proposed experimental setup can be extended to a much more general and complex system, both in modeling as well as control design leading to a new benchmark problem in the control system.

Exit time asymptotics for dynamical systems with fast random switching near an unstable equilibrium

We consider the exit problem for a one-dimensional system with random switching near an unstable equilibrium point of the averaged drift. In the infinite switching rate limit, we show that the exit time satisfies a limit theorem with a logarithmic deterministic term and a random correction converging in distribution. Thus, this setting is in the universality class of the unstable equilibrium exit under small white-noise perturbations.

The Chaotic Influence of Convinced Elements on Routing Collective Opinion and Appearance of Butterfly Effect

The study touches on the impact of the presence of convinced opinion for some of intelligent agents during the voting on formation the final collective opinion in a community G consists of autonomous agents, every intelligent agent have his opinion 0 or 1 On a thoughtful topic, between these agents happening an interaction[1], by entering into a collective debate[2], this interaction we will call 'voting', especially in the community, when the number of elements that have opinion 0, convergent compared with the number of elements that have different opinion 1 during the voting where this state of society is close to an unstable equilibrium point, this state will change during voting process, to close to one of two stability points 0 or 1 according to a previous study, presence convinced opinion intelligent agents, would later lead to a chaotic change in the final result of the voting[3], this chaotic change is called the “Butterfly Effect”[4,5], this influence can occur in many fields, market movements and economics, social life, mathematical problems, technological topics, movement of a driverless or ;