This paper advances hypercomplex neural-network theory by introducing Dual-Split-Quaternion (DSQ) algebra for control and synchronization. Whereas standard quaternions model only three-dimensional Euclidean rotations, DSQ algebra combines a nilpotent dual unit (ϵ2 = 0) with split-quaternion bases (i2 = −1, j2 = 1, k2 = 1), making it suitable for coupled non-Euclidean rotational and translational representations. These algebraic properties also introduce zero-divisors, non-compactness, and nilpotent degeneracy, which invalidate conventional Lyapunov metric constructions. To address this issue, we establish a rigorous framework for Fractional-Order Dual-Split-Quaternion Valued Memristive Neural Networks (FODSQVMNNs) with constant transmission delays, and we prove an absolute composite norm inequality in DSQ space. Using Filippov differential inclusions for memristive switching, we synthesize an event-triggered fixedtime sliding-mode controller. The resulting analysis guarantees synchronization within a computable fixed time Tmax that is independent of initial conditions. Comparative tables and embedded mathematical simulations validate the theoretical and computational advantages of the proposed framework. Finally, synchronized non-compact 16-dimensional hyperchaotic attractors generated from coupled DSQ states are leveraged to construct a 6D relativistic color-image encryption scheme with strong resistance to statistical and differential attacks.
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- Article
Global Fixed-Time Synchronization of Fractional-Order Dual-Split-Quaternion Valued Memristive Neural Networks: Full Proofs and 6D Relativistic Image Encryption
Author Information
Received: 01 Apr 2026 | Revised: 28 May 2026 | Accepted: 29 May 2026 | Published: 07 Jul 2026
Abstract
Keywords
dual-split-quaternion | memristive neural networks | fractional calculus | fixed-time synchronization | filippov inclusion | relativistic encryption
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