  This work focuses on the detection of the buckling phenomena and bifurcation
analysis of the parametric Von K\'arm\'an plate equations based on reduced
order methods and spectral analysis. The computational complexity - due to the
fourth order derivative terms, the non-linearity and the parameter dependence -
provides an interesting benchmark to test the importance of the computational
reduction strategies, during the construction of the bifurcation diagram by
varying the parameter(s). To this end, together the state equations, we carry
out also an analysis of the linearized eigenvalue problem, that allows us to
better understand the physical behaviour near the bifurcation points, where we
lose the uniqueness of solution.
