Figure 1.

Stochastic destabilization by internal white-noise perturbation. The Jacobian matrix is $A=\left(\begin{array}{cc}-1& 100\\ -1& -1\end{array}\right)$. We have that $1/\parallel {\hat{A}}^{-1}\parallel \simeq 0.04$, so that according to (3.8) fluctuations $\mathcal{P}$ with intensity $\parallel \mathcal{P}\parallel \ge 0.04$ affecting the matrix A can destabilize the equilibrium. We show a realization of the process dX_t=(A dt+σP dW_t)X_t with $P=\left(\begin{array}{cc}-0.07& -0.27\\ -0.92& 0.37\end{array}\right)$ and ∥P∥=1. In the rightmost panel, the variance σ²=0.04 is large enough to show premises of destabilization. Recall that for this matrix the real stability radius was independent of M, with ${\mathcal{S}}_{\mathrm{STR}}^{\mathrm{\Re }(c)}(A)=1$.