once you have decided the topic as " Nonlinear system identification". you need to research on the source of the nonlinearity and then plane to cover as mach you can of it in your phd.

I recommend you read the following book:

Worden, K.

Nonlinearity In Structural Dynamics : Detection, Identification, And Modelling

Institute of Physics, c2001

This topic is very general as the nature and expression of non-linearities that are significative in mechanical structures is almost problem dependent.

You thus need to assume a non-linearity model prior to performing identification, and it seems that a PhD can be spent on developing ad-hoc procedures for a specific non linearity type.

I suggest checking some recent work from Kerschen et al, about non linear system identification applied to structural dynamics, and also an article targetting viscoelasticity to increase your insight in the domain.

Kerschen, Gaetan, et al. "Past, present and future of nonlinear system identification in structural dynamics." Mechanical Systems and Signal Processing 20.3 (2006): 505-592.

Noël, Jean-Philippe, Ludovic Renson, and Gaëtan Kerschen. "Experimental identification of the complex dynamics of a strongly nonlinear spacecraft structure." Proceedings of the ASME IDETC/CIE 2013 (2013).

Jrad, Hanen, et al. "Experimental Characterization, Modeling and Parametric Identification of the Non-Linear Dynamic Behavior of Viscoelastic Components." European Journal of Mechanics-A/Solids (2013).