Chaotic systems without equilibrium are of interest because they are the systems with hidden attractors. A nonequilibrium system with chaos is introduced in this work. Chaotic behavior of the system is verified by phase portraits, Lyapunov exponents, and entropy. We have implemented a real electronic circuit of the system and reported experimental results. By using this new chaotic system, we have constructed S-boxes which are applied to propose a novel image encryption algorithm. In the designed encryption algorithm, three S-boxes with strong cryptographic properties are used for the sub-byte operation. Particularly, the S-box for the sub-byte process is selected randomly. In addition, performance analyses of S-boxes and security analyses of the encryption processes have been presented.

This article studies synchronization issues for a class of discrete-time fractional-order quaternion-valued uncertain neural networks (DFQUNNs) using nonseparation method. First, based on the theory of discrete-time fractional calculus and quaternion properties, two equalities on the nabla Laplace transform and nabla sum are strictly proved, whereafter three Caputo difference inequalities are rigorously demonstrated. Next, based on our established inequalities and equalities, some simple and verifiable quasi-synchronization criteria are derived under the quaternion-valued nonlinear controller, and complete synchronization is achieved using quaternion-valued adaptive controller. Finally, numerical simulations are presented to substantiate the validity of derived results.

In this paper, we investigate the dynamics of a fractional order chaotic map corresponding to a recently developed standard map that exhibits a chaotic behavior with no fixed point. This is the first study to explore a fractional chaotic map without a fixed point. In our investigation, we use phase plots and bifurcation diagrams to examine the dynamics of the fractional map and assess the effect of varying the fractional order. We also use the approximate entropy measure to quantify the level of chaos in the fractional map. In addition, we propose a one-dimensional stabilization controller and establish its asymptotic convergence by means of the linearization method.

A map without equilibrium has been proposed and studied in this paper. The proposed map has no fixed point and exhibits chaos. We have investigated its dynamics and shown its chaotic behavior using tools such as return map, bifurcation diagram and Lyapunov exponents’ diagram. Entropy of this new map has been calculated. Using an open micro-controller platform, the map is implemented, and experimental observation is presented. In addition, two control schemes have been proposed to stabilize and synchronize the chaotic map.

The load frequency control (LFC) system, a critical component maintaining frequency stability in the smart grid, is vulnerable to invisible false data injection attacks (FDIAs). These FDIAs can get behind the bad data detection (BDD) system and cause significant damage to the smart grid. Firstly, This paper proposes a detection and defense model against unobservable FDIAs in the LFC system based on the combination of the attack-detection evolutionary game (AEG) model and Kalman filtering (KF) algorithm. Then, Support vector machines (SVM) and K-Nearest neighbor (KNN) are two detection algorithms of the AEG model that are trained by gathering historical data of frequency deviation, tie-line power deviation, and active power load deviation in the smart grid responding to two different forms of FDIAs. The optimal detection algorithm is provided by analyzing the evolution of the equilibrium point of the game. Finally, a mitigation method based on the KF algorithm is proposed, in which the optimal control signal is estimated and issued to the LFC system in order to restore the system frequency stability. Simulation results on a two-area interconnected power system demonstrate the effectiveness of the proposed detection and defense strategies.