Predefined-time synchronization of 5D Hindmarsh–Rose neuron networks via backstepping design and application in secure communication

In this paper, the fast synchronization problem of 5D Hindmarsh–Rose neuron networks is studied. Firstly, the global predefined-time stability of a class of nonlinear dynamical systems is investigated under the complete beta function. Then an active controller via backstepping design is proposed to achieve predefined-time synchronization of two 5D Hindmarsh–Rose neuron networks in which the synchronization time of each state variable of the master-slave 5D Hindmarsh–Rose neuron networks is different and can be defined in advance, respectively. To show the applicability of the obtained theoretical results, the designed predefined-time backstepping controller is applied to secure communication to realize asynchronous communication of multiple different messages. Three numerical simulations are provided to validate the theoretical results.