# Instability Induced by Random Background Noise in a Delay Model of Landslide Dynamics
## Appendix A  
We shall study a system of units modeled by the following set of stochastic delay differential equations:  
\[\begin{array}{l}dx_{i}(t)=y_{i}(t)dt\\ dy_{i}(t)=-\big{[}a(V+y_{i}(t))^{3}-b(V+y_{i}(t))^{2}+c(V+y_{i}(t))\big{]}dt+ \\ (aV^{3}-bV^{2}+cV)dt+\sum_{j=1}^{N}k_{1}\big{(}x_{j}(t-\tau)-x_{i}(t)\big{)}dt+ \sqrt{2D}dW_{i}\end{array} \tag{1}\]  
i.e.,  
\[\begin{array}{l}dx_{i}(t)=y_{i}(t)dt\\ dy_{i}(t)=-\big{[}3 aV^{2}\mathbf{y}_{i}(t)-3a\mathbf{V}\mathbf{y}_{i}^{2}(t)-ay _{i}^{3}(t)+2 bVy_{i}(t)+by_{i}^{2}(t)-cy_{i}(t)\big{]}dt+\\ +\sum_{j=1}^{N}k_{1}\big{(}x_{j}(t-\tau)-x_{i}(t)\big{)}dt+\sqrt{2D}dW_{i}\end{array} \tag{2}\]  
where \(a=3.2\), \(b=7.2\), \(c=4.8\), \(V=0.2\) are parameters and \(x_{i}=0.001\), \(y_{i}=0.001\), are initial conditions.  
Now we shall perform the cumulant analysis of a system of above mentioned globally coupled units in the thermodynamics limit of an infinitely large ensemble, N\(\rightarrow\infty\). In that case, the set of stochastic delay differential equations becomes:  
\[\begin{array}{l}dx_{i}(t)=y_{i}(t)dt\\ dy_{i}(t)=-\big{[}3 aV^{2}y_{i}(t)-3 aV\mathbf{y}_{i}^{2}(t)-ay_{i}^{3}(t)+2 bVy _{i}(t)+by_{i}^{2}(t)-cy_{i}(t)\big{]}dt+\\ k\langle x(t-\tau)-x_{i}(t)\rangle dt+\sqrt{2D}dW_{i}\end{array} \tag{3}\]  
where k = k\({}_{1}\)N, and \(\langle x(t-\tau)\rangle=\lim\limits_{N\rightarrow\infty}\frac{1}{N}\sum_{i= 1}^{N}x_{i}(t-\tau)\).  
Next, we introduce deviations from the mean-field \(\langle x(t)\rangle=\lim\limits_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^{N} x_{i}(t)\) and \(\langle y(t)\rangle=\lim\limits_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^{N}y_{ i}(t)\), for each element: \(n_{x}(t)=\langle x(t)\rangle-x_{i}(t),n_{y}(t)=\langle y(t)\rangle-y_{i}(t)\). We assume these fluctuations are Gaussian and statistically independent in different elements.  
A set of moments known as cumulants or Thiele semi-invariants has an important property that all vanish in the Gaussian case. Therefore, we introduce the following notation for the first and second-order cumulants: (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 and (c) the cross-cumulant \(U(t)=\big{\langle}n_{x}n_{y}\big{\rangle}\).  
From cumulant analysis, one can find that:\[\begin{array}{l}\langle y^{2}\rangle=\frac{1}{N}\sum\limits_{i=1}^{N}y_{i}^{2}=s_ {y}+m_{y}^{2}\\ \langle y^{3}\rangle=\frac{1}{N}\sum\limits_{i=1}^{N}y_{i}^{3}=m_{y}^{3}+3m_{y }s_{y}\\ \langle y^{4}\rangle=\frac{1}{N}\sum\limits_{i=1}^{N}y_{i}^{4}=m_{y}^{4}+6m_{ y}^{2}s_{y}+3s_{y}^{2}\\ \langle xy\rangle=\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}y_{i}=U+m_{x}m_{y}\\ \langle xy^{2}\rangle=\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}y_{i}^{2}=m_{x}s_{ y}+m_{x}m_{y}^{2}+2m_{y}U\\ \langle xy^{3}\rangle=\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}^{3}y_{i}=3s_{y}U+3 m_{y}^{2}U+m_{x}m_{y}^{3}+3m_{y}m_{x}s_{y}\end{array} \tag{10}\]  
From Equations (10) and (10) and by considering [PERSON]'s chain rule it follows:  
\[\dot{m}_{x}(t)=m_{y}(t) \tag{11}\]  
\[\begin{array}{l}\dot{m}_{y}(t)=\big{(}-3 aV^{2}+2 bV-c\big{)}m_{y}(t)+(b-3 aV) \Big{(}s_{y}(