Transient growth induces unexpected deterministic spatial patterns in the Turing process

Turing models are often invoked to explain spatial pattern formation in a number of physical, chemical and biological processes. Pattern occurrence is generally investigated through a classical eigenvalue analysis, which evaluates the asymptotic stability of the homogeneous state of the system. Here we show that deterministic patterns may emerge in a Turing model even when the homogeneous state is stable. In fact, the non-normality of the eigenvectors is able to generate transient (long-lasting) patterns even in the region of the parameter space where the dynamical system is asymptotically stable (i.e., the eigenvalues are negative). Moreover, non-normality-induced patterns usually display an interesting multiscale structure that can be investigated analyticall