 Solution

The matrix $N$ is a skew-symmetric matrix, so $x^{T}Nx=0$ for real $x$. Therefore the energy-neutral rotational nonlinearity contributes nothing directly to the energy balance:The nonlinear term rotates the state without changing its Euclidean norm. It can still change the orientation, and thereby affect later energy growth through the linear term. Thus energy cancellation does not mean the nonlinear and linear trajectories have identical energy histories.