In this paper, based on a three-dimensional Bao system, a memristor-based hyper-chaotic Bao-like system is successfully constructed, and a simulated equivalent circuit is designed, which is used to verify the chaotic behaviors of the system. Meanwhile, a control method called periodically intermittent control with variable control width is proposed. The control width sequence in the proposed method is not only variable, but also monotonically decreasing, and the method can effectively stabilize most existing nonlinear systems. Moreover, the memristor-based hyper-chaotic Bao-like system is controlled by combining the proposed method with the Lyapunov stability principle. Finally, we should that the proposed method can effectively control and stabilize not only the proposed hyper-chaotic system, but also the Chua’s oscillator.

So many real life problems ranging from medicine, agriculture, biology and finance are modelled by nonlinear systems. In this case, a chaotic nonlinear system is considered and, as opposed to solving Linear Matrix Inequality (LMI), which is the usual approach but cumbersome, a completely different approach was used. In some other cases, the computation of singular value of matrix was used but the method in this study needs not such. In addition, most models, if not all, concentrate on finding a control matrix J under some sufficient conditions. The problem is that only one such matrix J is provided. In reality, the actual control quantity may have a little deviation from the theoretical J. Hence, the study in this paper provides a set of infinite uncertain matrices ${\mathbb{J}}_{\alpha }$ which are able to adapt to control the system under uncertain conditions. It turns out that this new method controls the system in shorter time with less computational complexities.

The higher-order Hermite–Gaussian (HG) modes exhibit complex spatial distributions and find a wide range of applications in fields such as quantum information processing, optical communications, and precision measurements. In recent years, the advancement of deep learning has emerged as an effective approach for generating higher-order HG modes. However, the traditional data-driven deep learning method necessitates a substantial amount of labeled data for training, entails a lengthy data acquisition process, and imposes stringent requirements on system stability. In practical applications, these methods are confronted with challenges such as the high cost of data labeling. This paper proposes a method that integrates a physical model with deep learning. By utilizing only a single intensity distribution of the target optical field and incorporating the physical model, the training of the neural network can be accomplished, thereby eliminating the dependency of traditional data-driven deep learning methods on large datasets. Experimental results demonstrate that, compared with the traditional data-driven deep learning method, the method proposed in this paper yields a smaller root mean squared error between the generated higher-order HG modes. The quality of the generated modes is higher, while the training time of the neural network is shorter, indicating greater efficiency. By incorporating the physical model into deep learning, this approach overcomes the limitations of traditional deep learning methods, offering a novel solution for applying deep learning in light field manipulation, quantum physics, and other related fields.