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.

大量的研究集中在混沌系统上，主要是因为它们在广泛的领域中引起了人们的兴趣。混沌系统的全面理解和同步在实际应用中起着至关重要的作用。为此，本研究研究了多稳态分数阶混沌系统。首先，描述了系统的一些动力学特征，并验证了系统的混沌行为。然后，同时计算了谱熵和谱最小熵，并显示了多稳定性现象。此外，提出了一种新的无抖动鲁棒滑模控制器与神经网络观测器的组合，用于分数阶系统的同步。利用神经网络估计器，可以获得系统的未知功能，并且完全考虑了干扰的影响。此外，基于李雅普诺夫稳定性定理，可以确定闭环系统的渐近稳定性。最后，将所提出的控制技术应用于分数阶系统。数值结果表明，在存在未知时变外部干扰的情况下，所提出的不确定系统控制方法无抖动且有效。