From localized spot to the formation of invaginated labyrinth structures in spatially extended systems

The stability of a circular localized spot with respect to azimuthal perturbations is studied in in a variational Swift-Hohenberg model equation. The conditions under which the circular shape undergoes an elliptical deformation that transform it into a rod shape structure are analyzed. As it elongates the rod-like structure exhibits a transversal instability that generates an invaginated labyrinth structure which invades all the space available.