Periodicity and fixed-time periodic synchronization of discontinuous delayed quaternion neural networks

This paper focuses on the issue of periodic orbits analysis for discontinuous quaternion neural networks (QNNs) with time delay. First, based on topological degree theorem of set-valued analysis, a sufficient condition is obtained to ensure the periodicity of discontinuous delayed QNNs via decomposition method. Second, a new fixed-time convergence lemma is proposed to study fixed-time periodic synchronization of discontinuous delayed QNNs. Different from the existing fixed-time convergence lemma, the constructed Lyapunov function (LF) is allowed to possess indefinite derivative for almost everywhere. Moreover, a switching control algorithm is designed to realize the drive-response periodic synchronization in a fixed time. Finally, two examples are presented to substantiate the main results.