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Instabilities of localized structures in dissipative systems with delayed feedback.

S V Gurevich¹, R Friedrich

¹Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, D-48149 Münster, Germany. gurevics@uni-muenster.de

Physical Review Letters

|February 7, 2013

Summary

We discovered novel behaviors in localized structures using delayed feedback in the Swift-Hohenberg equation. Varying delay time and feedback strength creates oscillons, soliton rings, labyrinth patterns, or moving structures.

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Area of Science:

Nonlinear dynamics
Pattern formation
Mathematical physics

Background:

The Swift-Hohenberg equation models pattern formation in various physical systems.
Delayed feedback introduces complex dynamics not present in the original equation.
Understanding localized structures is crucial for predicting system behavior.

Purpose of the Study:

To investigate the novel behaviors of solitary localized structures in a real Swift-Hohenberg equation with delayed feedback.
To analyze the impact of varying delay time and feedback strength on structure formation.
To characterize the bifurcations leading to spontaneous motion of these structures.

Main Methods:

Numerical simulations of the Swift-Hohenberg equation with delayed feedback.
Bifurcation analysis of the delayed system.
Derivation of order parameter equations near bifurcation points.
Analysis of the normal form for spontaneous motion.

Main Results:

Nontrivial instabilities arise from variations in the product of delay time and feedback strength.
Diverse patterns emerge, including oscillons, soliton rings, labyrinth patterns, and moving structures.
A normal form for spontaneous motion was derived, showing motion without shape change at the lowest order.

Conclusions:

Delayed feedback in the Swift-Hohenberg equation leads to a rich variety of localized structure behaviors.
Bifurcation analysis provides a framework for understanding the transition to moving structures.
The derived normal form accurately describes the onset of spontaneous motion.