Orientation interval for transient energy growth (${\rho }^{\pm }=2R(1\pm \sqrt{1-R^{-2}})$)

For $L=\left(\begin{array}{cc}-1/R& 1\\ 0& -1/(4R)\end{array}\right)$ and $R>1$, write $\rho =x_2/x_1$. The energy derivative is $x_1^2[-1/R+\rho -{\rho }^2/(4R)]$, positive exactly between the displayed orientations. Linear trajectories cross the sector from ${\rho }^{+}$ to ${\rho }^{-}$ because $\dot{\rho }=\rho [3/(4R)-\rho ]$. An energy-neutral rotational nonlinearity can reverse the crossing direction; the growth sector itself remains the same.