Projective and functional projective synchronizations in quaternion-valued inertial neural networks with interaction term through novel generalized time-varying delay lemma and one-norm method: Application in secure communication.

Abstract

This study addresses projective synchronization (PS) and function projective synchronization (FPS) in quaternion-valued inertial neural networks (QVINNs) with generalized delays. A generalized delay-based lemma is proposed to establish synchronization criteria. Effective controllers are designed to guarantee PS and FPS between the drive and response systems. Using one-norm technique, sufficient conditions are derived for the reduced system. Two main theorems, Theorems 1 and 2, are presented. Theorem 1 has employed a complex Lyapunov function in the non-reduced case, while the Theorem 2 has adopted a simpler one-norm-based Lyapunov approach. The proposed framework reduces computational complexity and simplifies the proof process. Additionally, an audio encryption scheme based on random permutation matrix and synchronization is developed, and the results are validated through numerical simulations, confirming the effectiveness of the proposed results.