Analisi di biforcazione per strutture spazialmente non homogenee nel modello di FitzHugh-Nagumo

The effects of diffusive processes on the emerging non homogeneous structures in the FitzHugh-Nagumo model are investigated. It is considered different configurations supported by the reaction term of the model and a linear and a weakly non linear analysis are performed in each case, to understand which kind of structures are stable or not. A Turing instability analysis is carried out and normal form techniques are used to prove the stability of standard squares, stripes and hexagonal patterns and, moreover, the existence of stable superlattice patterns in presence of three homogeneous equilibria are shown. It is also considered oscillatory dynamics proving that oscillating patterns arise close to the Turing-Hopf bifurcation. A particular attention is given to excitable dynamics: in this case cross diffusion is necessary for the onset of the Turing instability and it is responsible for the existence of oscillating solutions in absence of Hopf bifurcation.