# Instability Induced by Random Background Noise in a Delay Model of Landslide Dynamics
## 3 Effect of Noise on Landslide Dynamics
(**c**) for the starting model (1), and (**d**) for the corresponding mean-field model. It is clear that both models exhibit qualitatively the same dynamics. While \(\tau\) and k are varied, other parameters are being held constant: V = 0.2, a = 3.2, b = \(-7.2\), c = 4.8, initial conditions: m\({}_{\rm x}\) = 0.001, m\({}_{\rm y}\) = 0.0001; s\({}_{\rm x}\) = s\({}_{\rm y}\) = 0.05, D = 0.0001, x = 0.001, y = 0.001. \[Fc(V)=3.2V^{3}-7.2V^{2}+4.8V \tag{2}\]1 Bifurcation diagram for the interaction of time delay \(\tau\) and spring stiffness \(k\) for model (3). While \(\tau\) and k are varied, other parameters are being held constant: V = 0.2, \(a\) = 3.2, \(b\) = \(-7.2\), \(c\) = 4.8, initial conditions: m\({}_{\rm x}\) = 0.001, m\({}_{\rm y}\) = 0.0001; s\({}_{\rm x}\) = s\({}_{\rm y}\) = 0.05, D = 0.0001. Velocity time series for points (\(k=10\), \(\tau=1\)) and b (\(k=10\), \(\tau=3\)), both for the models (1) and (3), are given in \[Fc(V)=3.2V^{3}-7.2V^{2}+4.8V \tag{2}\]2  
One can easily show that the dynamics of the starting stochastic model (1) and the mean-field model (3) before and after the bifurcation curve are qualitatively similar, as it is shown in \[Fc(V)=3.2V^{3}-7.2V^{2}+4.8V \tag{2}\]3 The first row shows the mean velocity of the starting stochastic system (1), Figure 8a, and the mean of the deterministic mean-field model (3), Figure 8b, before the bifurcation curve shown in Figure 7, for \(k=10\) and \(\tau=1\) (point a). It is clear that both systems are in an equilibrium state. The second row shows the mean velocity of the starting stochastic system (1), Figure 8c, and the mean of the deterministic mean-field model (3), Figure 8d, after the bifurcation curve shown in Figure 7, for \(k=10\) and \(\tau=3\) (point b). Both systems exhibit periodic oscillations. Regarding the effect of noise on landslide dynamics, further analyses imply the following. Figure 9 shows the effect of different noise intensities on the position of the bifurcation curve in the \(k\)-\(\tau\) diagram. Landslide dynamics before the bifurcation curves are considered to be stable (creeping state), while landslide dynamics after the bifurcation curves are considered to be unstable. It seems there is a positive effect of random background noise on landslide stability. In particular, with the increase of the intensity of random noise from \(10^{-4}\) to \(10^{-2}\), the Hopf bifurcation curve \"moves\" further, i.e., higher values of time delay or spring stiffness are needed for the instability to occur. This effect is pronounced both for the low and high values of time delay and spring stiffness, as shown in \[Fc(V)=3.2V^{3}-7.2V^{2}+4.8V \tag{2}\]4 Such an effect for high values of time delay (3.5-5.0) and relatively low values of spring stiffness is shown in Figure 9a. These conditions are expected in strongly weathered rock masses, with loose connections between the actual moving parts of the unstable slope, where higher values of delayed interactions are expected. It is clearly seen that for constant spring stiffness and time delay, solely by increasing the noise intensity, the system under study becomes stable. The effect of noise for low values of time delay (0.75-0.80) and high values of spring stiffness is shown in Figure 9b. Such a regime corresponds to rock mass which is just slightly weathered with low deformability, where low values of delayed interaction are expected. One could see that increase of \(D\) for constant \(k\) and \(\tau\) leads to the stabilization of landslide dynamics. Additionally, vice versa, the reduction of \(D\) destabilizes the dynamics of the landslide. Moreover, this effect is pronounced between \(10^{-4}\) and \(10^{-3}\), while the further effect of \(D\) is significantly lower. This is a rather unexpected result implying a positive role of noise on slope stability compared to the influence of delay interaction or coupling stiffness.