When projecting a vector in Hilbert space into its (closed?) subspace, its best approximation is its Fourier series. The technique has been using in many traditional problems (heat, wave, Schrödinger) and in other low dimensional dynamical systems by finding $\lambda$ in the characteristic polynomial $det(A-\lambda I)$.
However, in general, are there any differences when applying this to any given dynamical system compared to the traditional ones? Not all systems have nice or symmetrical equations, and they may involve more variables/higher dimensions, and I think it might be stuck to find the characteristic polynomial. Is there ever such a thing, and how to solve it?