# Instability Induced by Random Background Noise in a Delay Model of Landslide Dynamics
## 3 Effect of Noise on Landslide Dynamics
Adopted friction law, as shown in Equation (2), adequately mimics the cubic friction law existing along the potential (and real) sliding surface, with the natural transition from velocity strengthening to velocity weakening regime. Equation (1) as a set of n stochastic delay differential equations is examined in the present paper using the mean-field approximation, by which one obtains the following mean-field model of the starting system (1):  
Figure 5: Examined an infinite chain of blocks on an inclined slope as a landslide model: k stands for the spring stiffness, while V denotes the constant background velocity. \[\dot{m}_{x}(t) =m_{y}(t) \tag{3}\] \[\dot{m}_{y}(t) =\big{(}-3 aV^{2}+2 bV-c\big{)}m_{y}(t)+(b-3 aV)\Big{(}s_{y}(t)+m_{y}^{2 }(t)\Big{)}-a\Big{(}m_{y}^{3}(t)+3m_{y}(t)s_{y}(t)\Big{)}\] \[\frac{1}{2}\dot{s}_{x}(t) =U(t)\] \[\frac{1}{2}\dot{s}_{y}(t) =s_{y}(t)\Big{[}\big{(}2 bV-3 aV^{2}-c\big{)}+2(b-3 aV)m_{y}(t)-3a \Big{(}m_{y}^{2}(t)+s_{y}(t)\Big{)}\Big{]}-kU(t)+D\] \[\dot{U}(t) =U(t)\Big{[}\big{(}2 bV-3 aV^{2}-c\big{)}+2(b-3 aV)m_{y}(t)-3a\Big{(} s_{y}(t)+m_{y}^{2}(t)\Big{)}\Big{]}+s_{y}(t)-ks_{x}(t)\]  
where the first and second-order cumulants are denoted in the following way: (a) the means \(m_{x}(t)=\langle x(t)\rangle,m_{x}(t-\tau)=\langle x(t-\tau)\rangle,m_{y}(t)= \langle y(t)\rangle\), (b) the mean square deviations \(s_{x}(t)=\big{\langle}n_{x}^{2}(t),s_{y}(t)=\Big{\langle}n_{y}^{2}(t)\Big{\rangle}\) and (c) the cross-cumulant \(U(t)=\big{\langle}n_{x}n_{y}\big{\rangle}\). Noise intensity is denoted with \(D\), \(V\) stands for the block velocity, while \(a\), \(b\), and \(c\) denote cubic friction parameters. Details of the derivation of model (3) are given in Appendix A. The justification for reducing the model of n units to a single mean-field model lies in the very nature of the specific landslide \"Zavoj\" itself, for which the recorded background seismic noise was also examined. In particular, the whole landslide body is composed of the weathered bedrock material (Figure 6), whose dynamics, therefore, could be analyzed as the behavior of the reduced mean-field model. Moreover, in the present paper, we are interested in the global stability of the landslide model. This is why we seek the transition between the global dynamical regimes, while small dynamical effects that may occur due to the dynamics of individual blocks are not of interest in the present paper. The analysis of the influence of random background noise on the dynamics of the landslide model (3) is conducted numerically, using the 4 th order Runge-Kutta numerical method, under the effect of the introduced time delay \(\tau\) and spring stiffness \(k\). Results of the conducted analysis indicate the occurrence of the direct supercritical Hopf bifurcation, with an increase in time delay and/or spring stiffness (Figure 7). Particularly, the observed dynamical system goes from an equilibrium state (represented as small constant background displacement, the so-called \"creep regime\") to regular periodic oscillations, which are considered unstable dynamical regimes. The trend of the bifurcation curve indicates the following: for the higher values of spring stiffness (or the low-deformable slopes), only the small values of displacement delay are required to trigger the instability. On the other hand, for low values of spring stiffness (high-deformable slopes), the increase of displacement delay has de facto no effect on the stability of the examined model. Figure 6: Characteristic engineering–geological cross-section along the left coast of the Zavoj dam. Figure 8: Time series of displacements before and after the bifurcation curve in \[Fc(V)=3.2V^{3}-7.2V^{2}+4.8V \tag{2}\]0 Upper diagrams show velocity time series for the parameters denoted by point a in Figure 7 (\(k=10\), \(\tau=1\)), (**a**) for the starting model (1), and (**b**) for the corresponding mean-field model. Lower diagrams show velocity time series for the parameters denoted by point b in Figure 7 (\(k=10\), \(\tau=3\)).