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Fractional order maps are a hot research topic; many new mathematical models are suitable for developing new applications in different areas of science and engineering. In this paper, a new class of a 2D fractional hyperchaotic map is introduced using the Caputo-like difference operator. The hyperchaotic map has no equilibrium and lines of equilibrium points, depending on the values of the system parameters. All of the chaotic attractors generated by the proposed fractional map are hidden. The system dynamics are analyzed via bifurcation diagrams, Lyapunov exponents, and phase portraits for different values of the fractional order. The results show that the fractional map has rich dynamical behavior, including hidden homogeneous multistability and offset boosting. The paper also illustrates a novel theorem, which assures that two hyperchaotic fractional discrete systems achieve synchronized dynamics using very simple linear control laws. Finally, the chaotic dynamics of the proposed system are stabilized at the origin via a suitable controller. [ABSTRACT FROM AUTHOR]

Since the variable-order fractional systems show more complex characteristics and more degrees of freedom due to time-varying fractional derivatives, we introduce a variable-order fractional Hopfield-like neural network in this paper. First, the properties and dynamical behavior of the system are studied. The variable-order derivative's effects on the system's behavior are investigated through the Lyapunov exponents and bifurcation diagram; an emerging Feigenbaum tree of period-4 bubble is observed, which appears with the creation and annihilation of periodic orbits. A general basin of attraction for the fractional-order neural network is presented, demonstrating that its dynamical behaviors are extremely sensitive to initial conditions resulting in different periodic orbits and chaotic attractors' coexistence. After that, an adaptive control scheme is proposed for the variable-order fractional system. Through Lyapunov theorem and Barbalat's Lemma, the system's convergence and stability under the proposed control scheme are proven. The main advantages of the proposed controller are its guaranteed stability, robustness against uncertainties, and simplicity. Finally, the synchronization results are presented. Numerical simulations show the excellent performance of the proposed controller for the variable-order fractional Hopfield-like neural network. [ABSTRACT FROM AUTHOR]

• In this paper, we present a new fractional-order hyperchaotic memristor oscillator. • The proposed system has been investigated through numerical simulations. • A novel robust adaptive controller is implemented to synchronize the system. • As an application, the system has been implemented for voice encryption. The present study proposes a new fractional-order hyperchaotic memristor oscillator. The proposed system is studied through numerical simulations and analyses, such as the Lyapunov exponents, bifurcation diagrams, and phase portraits. Then, using the sliding mode concept, a robust adaptive control scheme is designed to synchronize the proposed system. The adaptation mechanism is implemented to estimate the unknown parameters of the slave system. Then, the output of the proposed adaptation mechanism is used for the control scheme. The stability of the closed-loop system is proven via a fractional version of the Lyapunov stability theorem and Barbalat's lemma. Numerical simulations of synchronization are shown to investigate the performance of the developed control technique on the uncertain fractional-order hyperchaotic memristor oscillator. Finally, as an engineering application, the proposed fractional-order system is implemented for voice encryption. In this regard, to show the appropriate performance of the proposed system for voice encryption, statistical characteristic of the encryption and decryption processes are performed through different methods including correlation, entropy, root mean square, and root sum of squares. [ABSTRACT FROM AUTHOR]