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The neighboring coordinates are strongly correlated in 3D images, but the encryption format of 3D images is not suitable for 2D image encryption. Thus, a novel compatible chaotic image encryption system based on checkers rules and shift register is proposed in this paper. Firstly, a discrete memristive hyperchaotic map(a discrete hyperchaotic map coupled with a memristor) is used to generate the chaotic sequence of the cryptosystem. The hyperchaotic map is analyzed through Phase diagrams, Lyapunov exponent spectrum(LEs), and bifurcation diagram. Secondly, the data format of a 3D image is converted to a 2D image data format. Then, the checkers rules are used to scramble the 2D image data format. Finally, the scrambled data and the chaotic sequence are converted to character types, and the scrambled data is further encrypted by the shift register principle. The 2D image and 3D image are tested and the results show that the compatibility encryption system has good performance, significantly reducing the correlation among the data. [ABSTRACT FROM AUTHOR]

In the present paper, for the first time, we propose a variable-order hyperchaotic system for information security. Firstly, we study the dynamical behaviors of a memristor oscillator through well-known numerical and analytical tools, such as the Lyapunov exponents, stability of equilibria, and bifurcation diagram. Then as an engineering application, a variable-order fractional version of the system is proposed for sound encryption. In comparison with integer and conventional constant fractional-order chaotic memristor oscillator, the proposed variable-order fractional system shows more complex characteristics and more degrees of freedom due to the existence of time-varying fractional derivatives. Thus, the proposed system is an appropriate choice for data transmission and information security. To illustrate the proper performance of the suggested system for encryption purposes, sound encryption is successfully performed, and its excellent results are demonstrated. The predictor-corrector method is utilized for numerical simulation. Then, a new type-2 fuzzy disturbance observer-based robust control is offered for synchronization of the variable-order hyperchaotic system. The stability and convergence of the disturbance estimator and closed-loop system are proven. Lastly, the synchronization results, which confirm the appropriate performance of the proposed method in the presence of the external disturbances, are demonstrated. [ABSTRACT FROM AUTHOR]

Nowadays, chaotic systems are attracting considerable attention due to their special features. The aim of this paper is to study a four-dimensional system that lacks any equilibria. Also, this non-equilibrium system has hidden attractors. Dynamical behaviors of the system have been investigated using well-known numerical simulations and analyses such as phase portraits, Poincaré maps, Lyapunov exponents, bifurcation diagrams, and basins of attraction. This way, the chaotic behavior in the system is verified. In addition, the chaotic behavior of the system has been studied by designing and implementing a real-time analog circuit. The circuitry implementation of the system is presented, and the dynamical behaviors of the system have been empirically demonstrated. To characterize the circuit an automatic routine through LabView has been developed in which by increasing a parameter of the system the bifurcation diagram has been obtained and depicted. A DAQ 6363 storing temporal series of one second to posteriory has been used to acquire the data and analyze such information. Experimental results clearly illustrate the chaotic behavior of the system. Also, the presented experimental results pave the way for using the proposed system in possible engineering applications such as data transmission and information security. [ABSTRACT FROM AUTHOR]

• The discrete-time mathematical model of the macroeconomic system is presented. • The system is studied through topological classification, bifurcation analysis, Lyapunov exponents, and manifold theory. • A fuzzy based-activation feedback controller is proposed for the synchronization of the system. Economic systems, due to their substantial effects on any society, are interesting research subject for a large family of researchers. Despite all attempts to study economic and financial systems, studies on discrete-time macroeconomic systems are rare. Hence, in the current study, we aim to investigate dynamical behavior and synchronization of these systems. At first, the discrete-time mathematical model of the macroeconomic system is presented. Then, the system is studied through topological classification, bifurcation analysis, Lyapunov exponents, and manifold theory, which are powerful tools in the investigation of nonlinear systems. This way, the features of the system are disclosed, and the existence of chaos in the system is shown. For the adequate performance of the economy, the economic systems are desired to operate in a unified manner. To this end, in the present research, a fuzzy based-activation feedback controller is proposed for the synchronization of the system. To enhance the celerity and accuracy of the proposed control for synchronization purposes, it is equipped with a fuzzy logic engine. Finally, the numerical simulations of the synchronization are presented and compared with those of a conventional activation feedback control. Numerical results verify that the proposed control technique can successfully push the states of the response system to the desired value. [ABSTRACT FROM AUTHOR]

• The effect of market confidence on a financial system from the perspective of fractional calculus. • Investigation of chaotic behavior of the system through numerical simulations and analyses, such as the Lyapunov exponents, bifurcation diagrams, and phase portraits. • Designing and implementing an analog circuit to prove the feasibility of the system in real-world applications. Modeling and analysis of financial systems have been interesting topics among researchers. The more precisely we know dynamic of systems, the better we can deal with them. This way, in this paper, we investigate the effect of market confidence on a financial system from the perspective of fractional calculus. Market confidence, which is a significant concern in economic systems, is considered, and its effects are comprehensively investigated. The system has been studied through numerical simulations and analyses, such as the Lyapunov exponents, bifurcation diagrams, and phase portrait. It is shown that the system enters chaos through experiencing a cascade of period doublings, and the existence of chaos is verified. Finally, an analog circuit of the chaotic system is designed and implemented to prove its feasibility in real-world applications. Also, through the circuit implementation, the effects of different factors on the behavior of the systems are investigated. [ABSTRACT FROM AUTHOR]

• In this paper, we introduce a fractional-order hyperchaotic economic system. Additionally, the transient chaos phenomenon is detected for lower fractional orders. • The dynamical behavior of the system is numerically investigated using bifurcations diagrams, basins of attraction, and Lyapunov exponents. • We propose an adaptive terminal sliding mode control with neural network estimator for finite-time stabilization and synchronization of the fractional-order system. Moreover, the effects of external disturbances are fully taken into account with neural network estimator. We introduce for the first time a fractional-order hyperchaotic economic system. In this system, chaos generation depends upon the value of fractional-order. For certain fractional orders, a sustained regime of chaos is obtained. Also, the transient chaos phenomenon is detected for lower fractional orders. The dynamical behavior of the system is numerically investigated using bifurcations diagrams, basins of attraction, and Lyapunov exponents. Next, an adaptive terminal sliding mode control (ATSMC) with neural network estimator for finite-time stabilization and synchronization of the fractional-order system has been proposed. The radial basis function (RBF) neural network is used to achieve the estimation of the unknown function of the system. Also, the effects of external disturbances are fully taken into account with neural network estimator. The weights of the RBF neural network are updated based on the appropriate adaptation law. Using the fractional version of the Lyapunov stability theorem, the finite-time convergence of the closed-loop system has been proven. Finally, the new control technique is used for control and synchronization of the fractional-order hyperchaotic economic system. Simulation results illustrate the effectiveness of the proposed control scheme for uncertain fractional-order systems in the presence of external disturbances. [ABSTRACT FROM AUTHOR]

Financial systems present nonlinear dynamics whilst in some conditions chaotic behaviours emerge, rendering the stability of these systems utterly volatile. In this novel study we investigate boom-bust cycles in stock markets with heterogeneous agent-based modeling, yet incorporating endogenously evolving fractions of market participants, particularly those switching between contrarian and fundamental strategies. The dynamical behavior of various market entry behaviours of market agents are studied via numerical simulations and analyses of phase portraits, bifurcation diagrams and Lyapunov exponents. If the system adjusts in a partial equilibrium of at least one positive Lyapunov exponent, we observe the emergence of chaotic dynamics. Overall, chaos emerges via endogenous boom-bust cycle dynamics, caused by time-varying fractions of switching strategies between contrarian versus fundamental trading strategies. Our results prove to be invaluable to different types of market participants, such as traders, arbitrageurs, investors, speculators and institutional market makers. • We investigate boom-bust cycles in financial markets with computational agent-based micro-modeling • Our novel approach incorporates evolving fractions of market entries switching between contrarian and fundamental strategies • Chaotic dynamics emerge endogenously due to time-varying fractions of heterogeneous market agents with adaptive expectations • Our scenario analysis uncovers chaotic dynamics that contribute to the observed boom-bust cycles in financial markets. • Our findings can be valuable for traders, arbitrageurs, investors, speculators and institutional market makers [ABSTRACT FROM AUTHOR]

• We introduce for the first time in the literature a fractional-calculus-based artificial macroeconomic model. • A new system of fractional difference-order equations is proposed for the Goodwin model. • Using bifurcation diagrams, phase portraits and Lyapunov exponents among other measures, the characteristics of the new system are depicted. • We implement extensive simulations via a new hardware laboratory realization set up. In this novel research, through dynamical analysis, we introduce for the first time a fractional-calculus based artificial macroeconomic model, actually implemented in the Laboratory via a new hardware set up. Firstly, we propose a new model of a discrete-time macroeconomic system where fractional derivatives are incorporated into the system of equations. Using well-known tools and methods, including bifurcation diagrams and Lyapunov exponents, the characteristics of the system are disclosed, and the importance of the fractional-order derivative in the modeling of the system is shown. After that, a laboratory hardware realization is also carried out for the proposed system that provides further insight and a better understanding of the properties of the system. For the hardware realization an Arduino Due™ is chosen in which possess two analog output pins. Experimental results conspicuously illustrate the chaotic behavior of the system. Through the results of the hardware realization, phase portraits and bifurcation diagram of the system are demonstrated, and the effects of the parameters and fractional derivatives are studied. We believe the presented study and its results pave the way for future studies on the incorporation of fractional calculus into macroeconomic models. [ABSTRACT FROM AUTHOR]