Instabilities of Localized Structures in Dissipative Systems with Delayed Feedback

We are interested in stability properties of solitary localized structures in
a real Swift-Hohenberg equation subjected to a delayed feedback. We shall show
that variation in the product of the delay time and the feedback strength
leads to nontrivial instabilities resulting in the formation of oscillons,
soliton rings, labyrinth patterns, or moving structures. We provide a
bifurcation analysis of the delayed system and derive a system of order
parameter equations explicitly describing the temporal behavior of the
localized structure in the vicinity of the bifurcation point. In addition, we
demonstrate that a normal form of the bifurcation, responsible for the
emergence of moving solitary structures, can be obtained and show that
spontaneous motion to the lowest order occurs without change of the shape.