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• Spectral entropy and spectral Min-Entropy are computed, and the phenomenon of multi-stability is shown. • A new chattering-free robust sliding mode controller with a neural network observer is proposed. • Based on the Lyapunov stability theorem, the asymptotical stability of the closed-loop system is confirmed. • Numerical results demonstrate the chattering-free and effective performance of the proposed control method. An immense body of research has focused on chaotic systems, mainly because of their interesting applications in a wide variety of fields. A comprehensive understanding and synchronization of chaotic systems play pivotal roles in practical applications. To this end, the present study investigates a multi-stable fractional-order chaotic system. Firstly, some dynamical features of the system are described, and the chaotic behaviour of the system is verified. Then, both spectral entropy and spectral Min-Entropy are computed, and the phenomenon of multi-stability is shown. Besides, the combination of a new chattering-free robust sliding mode controller with a neural network observer is proposed for the synchronization of the fractional-order system. With the neural network estimator, unknown functions of the system are obtained, and the effects of disturbances are completely taken into account. Also, based on the Lyapunov stability theorem, the asymptotical stability of the closed-loop system is confirmed. Lastly, the proposed control technique is applied to the fractional-order system. Numerical results demonstrate the chattering-free and effective performance of the proposed control method for uncertain systems in the presence of unknown time-varying external disturbances. [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]