## 4 Dynamical Analysis and Simulation Results
### Incommensurate Order Case  
Combined analysis of the bifurcation diagrams (Figure 5a,b) and the largest Lyapunov exponent plot (Figure 5c) reveals the complex dynamics of the incommensurate System (36) as the fractional order parameter \(\eta_{1}\) is varied, with \(\eta_{2}\) fixed at 0.86. For small values of \(\eta_{1}\) in the range \([0.25,0.55]\), the system displays chaos, as indicated by the dense and irregular patterns in the bifurcation diagrams. This chaotic regime is further confirmed by the MLE plot, where the exponent remains positive, signifying sensitive dependence on initial  
Figure 3: (a,b) Bifurcation charts of the commensurate system (12) as the fractional order parameter \(\eta\) varies. (c) The associated MLE.  
conditions. As \(\eta_{1}\) increases beyond approximately 0.55, the system transitions from chaos to periodic behavior through a period-doubling route, with distinct branches emerging in the bifurcation diagrams. This transition is also evident in the MLE plot Figure 5c, where the exponent gradually decreases and becomes negative around \(\eta_{1}=0.7\), indicating the system's stabilization. For \(\eta_{1}>0.7\), the system enters a periodic or steady-state regime, characterized by well-defined trajectories.  
Figure 6 provides a comprehensive view of the dynamic behavior of an incommensurate system as the fractional order \(\eta_{2}\) is varied, while \(\eta_{1}\) is held constant at 0.68. The bifurcation diagrams in Figure 6a,b depict the system's response to changes in \(\eta_{2}\), showcasing a rich tapestry of periodic windows interspersed with chaotic regions. This intricate pattern suggests a high sensitivity to variations in \(\eta_{2}\), implying that even small changes in this parameter can significantly alter the system's long-term behavior. Further evidence of this sensitivity is provided by the Maximum Lyapunov Exponent plot in Figure 6c. The MLE, a crucial indicator of chaoticity, fluctuates between positive and negative values. Positive MLE values signify chaotic behavior, marked by sensitivity to initial conditions, while negative values indicate periodic or stable behavior. The observed fluctuations in the MLE confirm the system's tendency to transition between these distinct dynamical regimes as \(\eta_{2}\) is adjusted. This dynamic interplay between order and chaos underscores the complex nature of the system and its intricate response to parameter variations. In essence, Figure 6 reveals a system with a delicate balance between order and chaos, while Figure 7 shows various phase portraits for different incommensurate orders. The bifurcation diagrams and the MLE plot collectively demonstrate that the system's behavior is remarkably sensitive to fractional order \(\eta_{2}\) changes. These findings highlight the significant role of the fractional parameter in governing the system's stability and complexity. Understanding the transition between chaotic and periodic behavior provides critical insights into computer virus propagation dynamics, aiding in the development of more effective cybersecurity strategies and mitigation techniques. In contrast, statistical-fitting approaches (e.g., [35]), while accurate in reproducing specific outbreak curves, do not capture the memory-dependent and chaotic behavior observed in real-world propagation scenarios. The model adds dynamic interpretability and flexibility through its fractional-order structure, allowing it to simulate a wider range of propagation patterns through parameter tuning.  
Figure 5: (a,b) Bifurcation charts of the incommensurate system (36) as the fractional order parameter \(\eta_{1}\) varies where \(\eta_{2}=0.86\). (c) The associated MLE.