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Instabilities of localized structures in dissipative systems with delayed feedback.
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.
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.
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