From Continuous Integer-Order to Fractional Discrete-Time: A New Computer Virus Model with Chaotic Dynamics

Computer viruses remain a persistent technological challenge in information security. They require mathematical frameworks that realistically capture their propagation in digital networks. Classical continuous-time, integer-order models often overlook two key aspects of cyber environments: their inherently discrete nature and the memory-dependent effects of networked interactions. In this work, we introduce a fractional-order discrete computer virus (FDCV) model, derived from a three-dimensional continuous integer-order formulation and reformulated into a two-dimensional fractional discrete framework. We analyze its rich dynamical behaviors under both commensurate and incommensurate fractional orders. Leveraging a comprehensive toolbox including bifurcation diagrams, Lyapunov spectra, phase portraits, the 0–1 test for chaos, spectral entropy, and C0 complexity measures, we demonstrate that the FDCV system exhibits persistent chaos and high dynamical complexity across broad parameter regimes. Our findings reveal that fractional-order discrete models not only enhance the dynamical richness compared to integer-order counterparts but also provide a more realistic representation of malware propagation. These insights advance the theoretical study of fractional discrete systems, supporting the development of potential technologies for cybersecurity modeling, detection, and prevention strategies.