Energy-neutral rotational nonlinearity ($x^{T}Nx=0,N^T=-N$)

A nonlinear term $f(|x|)Nx$ with real skew-symmetric matrix $N$ contributes zero to the derivative of the quadratic energy $|x{|}^2/2$. It changes orientation rather than magnitude directly. If a separate linear generator is a non-normal matrix, that reorientation can still alter later transient growth, so energy cancellation alone does not make the nonlinear evolution equivalent to the linear one.