Fractional-Order Diffusion of Complex Turing Pattern Formations of Malware Nodes in Cyber-Physical Systems

Motivated by the rapid rise of cyber-physical systems (CPSs) and the increasing activity of cyber attackers, this paper investigates the stability of a class of malware systems in CPSs involving cross-diffusion and fractional diffusion. An innovative application of Turing patterns is employed to intuitively simulate the spatial distribution changes of malware nodes in CPSs. By using weakly nonlinear analysis, the amplitude equations near the critical point of Turing bifurcation is derived. Subsequently, formation stability analysis is conducted to determine the types of Turing patterns and the thresholds for various modes. Numerical simulations reveal that reaction-diffusion effects can lead to distinct spatial patterns in the distribution of malicious nodes, including hexagonal, mixed, and stripe patterns. Furthermore, fractional-order diffusion significantly influences the density distribution of malware nodes, which has a crucial impact on system security. This study offers a novel perspective on the propagation mechanisms of malware in CPSs, contributing to enhanced insights in the field of cyber security. The findings provide valuable guidance for predicting and controlling malware propagation in CPSs, thereby enhancing the robustness and security of these systems against cyber threats.