The Lyapunov exponent of a randomly switched pair of stable planar systems is unimodal in the switching rate

Let $X_t$ solve $\dot X_t=A_{I_t}X_t$ in the plane, where $A_0=\bigl(\begin{smallmatrix}-\alpha&1\\0&-\alpha\end{smallmatrix}\bigr)$, $A_1=\bigl(\begin{smallmatrix}-\alpha&0\\-1&-\alpha\end{smallmatrix}\bigr)$ and $I_t$ switches between $0$ and $1$ at rate $r$. Both matrices are Hurwitz, yet for small $\alpha$ the system blows up for an intermediate range of rates. The top Lyapunov exponent has the form $G(r)-\alpha$, and Malrieu conjectured that $G$ increases and then decreases, with a single maximum of height about $0.2$, and tends to zero at both ends. We prove this. The maximum is attained at a single $r_*\in(0.226,0.235)$, $G(r)=r\log(1/r)\,(1+O(1/\log(1/r)))$ as $r\to0$, and $G(r)\le 1/(2r)+(2/\pi)K_1(r)$ for all $r$. Consequently the set of rates for which the system is unstable is an open interval when $\alpha<G(r_*)$ and is empty otherwise. The proof writes $G$ as a ratio of two one-dimensional integrals of an explicit kernel, settles $r\le 0.01$ and $r\ge5$ with elementary and Bessel-function bounds, and covers $[0.01,100]$ by interval arithmetic.