Solution
= Solution
The eigenvalues of $L$ are
$$
s_\pm=-\varepsilon\pm\sqrt{\beta^2-1}.
$$
Since $\beta<1$, the origin is linearly stable exactly when the determinant is positive:
$$
\boxed{1+\varepsilon^2-\beta^2>0},
$$
or $|\beta|<\sqrt{1+\varepsilon^2}$.