Instabilities of localized structures in dissipative systems with delayed feedback

TL;DR

This paper investigates how delayed feedback induces various instabilities in localized structures within a dissipative system, leading to complex patterns and motion, supported by bifurcation analysis and order parameter equations.

Contribution

It introduces a bifurcation analysis of localized structures under delayed feedback and derives explicit order parameter equations describing their dynamics near bifurcation points.

Findings

Delayed feedback causes instabilities leading to oscillons, rings, labyrinths, or moving structures.

Bifurcation analysis reveals the conditions for different pattern formations.

Normal form analysis shows spontaneous motion without shape change.

Abstract

We report on a novel behavior 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. 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.