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.

In this paper, some properties of a BPFR were studied. Also, level sets of BPFRs were studied with their properties. Some results were given and, in some cases, examples and counter-examples were constructed. In particular, we have shown that if a BPFR Q is a subset of another BPFR ρ , ( s, t )-level (or strong level) subsets of Q are classical subsets of the ( s, t )-level (or strong level) subsets of ρ and that the strong level subsets of a BPFR is a subset of the ordinary level subset of the same among others. By these, we have improved on some of the results in the previous work and have laid the right foundation for further works on these concepts.