I've been looking at the marginal Hilbert Spectra of both the simple harmonic oscillator and the anharmonic Morse oscillator. I have found that while the SHO has a marginal spectrum with a single and roughly lorentzian band, the corresponding Morse oscillator splits into two peaks, and the spectrum 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.