Detection of wobbling sound with rising notes
I would start with following set of parameters:


MFCC's (I that know you tried it, but stay with me) without static energy (1'st coefficient)
Some descriptors from MPEG-7, like: Spectral Flatness, Harmonicity, Fundamental frequency, Spectral Spread, etc. You can find out more about them here: click or in this great book: MPEG-7 Audio and Beyond


You should also include $\Delta$'s and $\Delta\Delta$'s of MFCC's and Spectral Flatness. They should really improve accuracy. You can also try to calculate these for other parameters. Having that, you can start with Gaussian Mixture Model or Support Vector Machine (I think that GMM will be suitable for that task). After training your model simply validate it on testing set (seems obvious but I've seen many projects where people tested their Machine Learning algorithms on training set with added noise or, what's even worse, same training data).

After this, you should perform Feature Optimisation. This is important part, as some of the features are irrelevant (i.e. Spectral Flatness for few first filter banks, while bird's chirp has higher frequencies). Nice publications with many algorithms for that purpose can be found here: click,   click.
Good luck!