# A Perturbation and Symmetry-Based Analysis of Mobile Malware Dynamics in Smartphone Networks
## 5 Numerical Methods and Implementation
Rotational invariance emerges near endemic equilibria \(\mathbf{x}^{*}=(M_{s}^{*},M_{i}^{*},M_{q}^{*},M_{r}^{*})\) in the \((M_{s},M_{i})\) subspace. For rotation operator \(R_{\theta}\) and small perturbations \(\mathbf{\epsilon}\):  
\[\mathbf{f}(\mathbf{x}^{*}+R_{\theta}\mathbf{\epsilon})\approx R_{\theta}\mathbf{f} (\mathbf{x}^{*}+\mathbf{\epsilon}) \tag{21}\]  
The spiral stability of endemic states becomes apparent when examining phase portraits, where trajectories from various initial angles converge uniformly toward \(\mathbf{x}^{*}\). The equilibrium coordinates  
\[ \begin{split} M_{s}^{*}&=\frac{\theta+\delta_{1}+ \delta_{2}}{\Pi}\\ M_{i}^{*}&=\frac{\delta_{1}(\mathcal{R}_{0}-1)}{ \Pi},\quad\mathcal{R}_{0}=\frac{\Pi\lambda}{\delta_{1}(\theta+\delta_{1}+ \delta_{2})}\\ M_{q}^{*}&=\frac{\theta_{1}}{\kappa}M_{i}^{*}, \quad\kappa=q+\delta_{1}+\delta_{3}\end{split}  \tag{22}\]  
contain the basic reproduction number \(\mathcal{R}_{0}\), highlighting how symmetry properties connect to epidemiological parameters. Approximate scaling symmetry is generated by  
\[X_{s}=\alpha t\partial_{t}+\sum_{k\in\{s,i,d,r\}}\beta_{k}M_{k}\partial_{M_{k}} \tag{23}\]  
Numerical optimization reveals optimal scaling weights \((\alpha,\beta_{s},\beta_{i},\beta_{q},\beta_{r})\approx\) (1.0, 1.0, 1.0, 1.0) with residual error \(\mathcal{O}(10^{-3})\), indicating nearly perfect dimensional homogeneity. The slight deviations originate from the constant recruitment term \(\lambda\) in (1), which breaks exact scalability. These symmetry properties collectively induce two conserved quantities through [PERSON]'s theorem:  
\[ \begin{split} I_{1}&=e^{-\alpha t}\prod_{k}M_{k}^{ \beta_{k}}\\ I_{2}&=(M_{s}-M_{s}^{*})^{2}+M_{i}^{2}\end{split}  \tag{24}\]  
where \(I_{1}\) corresponds to scaling symmetry and \(I_{2}\) to rotational invariance. The preservation of these quantities along trajectories provides model reduction opportunities: the 4D system can be analyzed through 2D symmetry-adapted coordinates without loss of dynamical information. Figure 5a: Phase space trajectory showing the relationship between susceptible mobile devices (\(M_{s}\)) and infected mobile devices (\(M_{i}\)). The blue solid line represents the original trajectory, while the red dashed line shows the same trajectory with a time shift. The perfect overlap confirms that solutions maintain their shape under time translation, with only their position in time changing. Figure 5b: Multiple trajectories with different initial conditions, all exhibiting the same fundamental dynamics but translated in the phase space. The black arrow indicates the direction of translation. This symmetry property is crucial for cybersecurity planning as it implies that the same containment strategies will be equally effective regardless of when an outbreak is detected and response measures are implemented. This visualization in Figure 6 displays trajectories in the susceptible (\(M_{s}\)), infected (\(M_{i}\)), and (\(M_{q}\)) phase spaces for the mobile malware model. Trajectories originating from an initial circle (dashed green) of perturbed states around the endemic equilibrium (red dot) demonstrate convergence towards this stable point, illustrating its role as an attractor in the system. Figure 7 examines symmetry aspects of the \(M_{s\!i\!qr}\) mobile malware model in the susceptible (\(M_{s}\))-infected (\(M_{i}\)) phase plane. For Figure 7a dynamics near disease-free equilibrium (DFE), the vector field and sample trajectories illustrate system convergence to the DFE (\(M_{s}=50.0\), \(M_{i}=0.0\)), indicating stability for sub-critical parameters (\(\mathcal{R}_{0}<1\)). The vertical line (\(M_{s}=50.0\)) is a visual reference. Figure 7b includes a test for approximate scaling symmetry. The original model trajectory (solid black) is compared against solutions de  
Figure 5: Mobile malware dynamics: time translation symmetry visualization. **(a)** Phase space plot of \(M_{s}\) vs. \(M_{i}\), showing the original (blue) and time-shifted (red dashed) trajectories.