I've been looking at the marginal Hilbert Spectra of both the simple harmonic oscillator and an anharmonic Morse oscillator. I have found that while the SHO has a marginal spectrum with a single and roughly lorentzian band shape, the corresponding Morse oscillator splits into two peaks, and the spectrum more closely resembles the probability density of a classical oscillator.
Looking at the instantaneous frequencies in the time domain indicates that the Morse oscillator frequencies change during the oscillation, and that as the oscillator spends more time at the turning points, this gives rise to the two distinct peaks.
However, it doesn't make physical sense to say that a single oscillating mode of a single oscillator has two distinct frequency components, and the equivalent Fourier spectrum gives a single peak roughly in between the two, corresponding roughly to the oscillators correct frequency.
This remains true when empirical mode decomposition (Hilbert-Huang Transform, HHT) is performed on the signal first so as to construct the marginal spectrum from intrinsic mode functions that should have well behaved Hilbert transforms.
I have tried doing this in both LabVIEW and MATLAB. What is the mathematical reason why this seemingly non-physical spectrum occurs for an anharmonic oscillation using the HHT? Alternatively does anyone know if might have made a mistake somewhere? If anyone knows of any relavent literature on the topic, that would help as well. As an aside if anyone could explain the origin of the Lorentzian in the harmonic case, that would help too.
I'm not familiar with Hilbert Spectra. But I'll point you at "Analysis and Design of Autonomous Microwave Circuits" chapter 3 "Bifurcation Analysis". There are stable and unstable branches in phase space which can cause an oscillator to jump to an alternate branch as conditions change. I'm not sure it's the same thing, but oscillators can definitely do weird things because they are non-linear. The Lorentzian spread looks like the plots in a lot of the graphs in that book, so "that's how it works" is about the best answer I can come up with for the moment.
I got the book to understand self injection oscillators. It's been a while since I looked at that stuff in detail, but I suspect you'll find useful information and good references too.