Direct approach on projective synchronization of inertial quaternion-valued neural networks with proportional delays

In this paper, inertial quaternion-valued neural networks with proportional delays are proposed. First, the projective synchronization of the addressed model is investigated by constructing a new Lyapunov functional with tuning parameters. Several criteria are derived to ensure the projective synchronization of quaternion-valued neural networks with proportional delays and inertial term. The most popular results of inertial quaternion-valued neural networks are based on the variable substitution approach, which reduces the original second-order system to a first-order system. The quaternion-valued neural networks can be separated into four equivalent real-valued neural networks, and this has made the process more complicated. In this paper, the authors deal with the non-separation approach and non-reduction order method for inertial quaternion-valued neural networks with proportional delays. In addition, two numerical simulation examples are given to validate the theoretical results obtained.