Spectra and pseudospectra in the evaluation of material stability in phase field schemes

We consider the dynamics of bodies with ‘active’ microstructure described by vector-valued phase fields. For waves with time-varying amplitude, the associated evolution equation involves a matrix that can be non-normal, depending on the constitutive choices adopted for the microstructural actions associated with the considered phase field. The occurrence of non-normality requires to look at the pseudospectrum of the considered matrix, namely the set of all possible eigenvalues of matrices in a $\varepsilon$-neighborhood of the matrix itself, because the eigenvalues of non-normal matrices can be very sensitive to small perturbations and therefore the spectral analysis alone would not be sufficient to distinguish with certainty between stable and unstable behavior. We develop the relevant analyses in the case of quasicrystals for which the values of some constitutive parameters are not known or are uncertain from an experimental point of view, a circumstance suggesting parametric analyses. We find circumstances in which the pseudospectra obtained by means of the so-called structured perturbations predict instability when, instead, the spectral analysis indicates stability.