The most general method is sum of squares based verification (based on Lyapunov theory). github.com/oxfordcontrol/SOSTOOLS/blob/SOSTOOLS400/README.md‌​. It doesn't rely on intuition, however the drawback is that the computation cannot scale to large sized systems.

ai.stackexchange.com/questions/27196/…

You have to prove 2 things: 1). The jacobian has a pair of purely imaginary e.values at bifurcation point. 2) The derivative of real part of this pair of eigenvalues w.r.t to parameter is non-zero at bifurcation point. From these two conditions, the Hopf bifurcation follows. See: "Elementary Stability and Bifurcation theory", Iooss and Joseph.

Reminds me of negative viscosty in certain flows, e.g. see : Słomka, Jonasz, and Jörn Dunkel. "Geometry-dependent viscosity reduction in sheared active fluids." Physical Review Fluids 2.4 (2017): 043102.

Yes the example is special case but you mentioned special cases in your question, e.g., Euler top/ Dihanibekov effect can occur in this special case. Re: center not being center in nonlinear system, well the whole phase space portrait is plotted in that video, so you can exactly see what happens. It remains center.

Regarding stability: since it is Hamiltonian system, each equilibria will either be a center or saddle type. Center is neutrally stable leading to precession, and saddle is unstable and will lead to tumbling

This is all very standard. The trajectories (in 3D phase space of 3 angular speeds about 3 principle axis) lie on the intersection of 2 ellipsoids that are surfaces of constant angular momentum and energy. See video here (starting at 13 min mark): youtube.com/watch?v=DPu6hb2HN_8

Can you add a bifurcation diagram ?

You are looking for Laplace transform of $Sin(\omega t)$ at $s= \alpha+i\omega$. Simply consult the Laplace transform table

When reducing from a very high degree of freedom model to a low degree one, often times the ignored degrees of freedom are modeled as noise. Infact, thats the origin of many commonly used SDEs.

people.math.wisc.edu/~thiffeault/talks/gfd2014_notes.pdf

@JochenGlueck yes, ofcourse I overlooked the semi-simple condition. You are right.

Spectrum of A (as in the given equation above) lies on the right-half plane plus the imaginary axis. If A has that property, then solution of the equation will be bounded for well-behaved forcing $f$, via use of variation of constants formula. These definitions regularly pop up in control theory literature.

A very readable reference is these lecture notes:arxiv.org/abs/1203.2344