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Updated: Jul 15, 2026
06:42
Magnetically Induced Rotating Rayleigh-Taylor Instability
Published on: March 3, 2017
Stability of localized solutions in a subcritically unstable pattern-forming system under a global delayed control
B Y Rubinstein1, A A Nepomnyashchy, A A Golovin
1Department of Mathematics, University of California, Davis, California 95616, USA.
Summary
Feedback control stabilizes localized patterns in systems with subcritical instability. Increasing delay causes oscillations and, eventually, solution blow-up, revealing complex dynamics under delayed feedback.
Area of Science:
- Nonlinear dynamics
- Pattern formation
- Control theory
Background:
- Subcritical instability in physical systems often leads to pattern formation.
- Feedback control is a key technique for manipulating system dynamics.
- Delayed feedback introduces complex behaviors, including oscillations and instabilities.
Purpose of the Study:
- Investigate the effect of delayed feedback control on spatially localized patterns.
- Analyze the stability and dynamics of these patterns under varying control parameters.
- Determine the conditions leading to pattern stabilization, oscillation, and blow-up.
Main Methods:
- Utilizing the globally controlled Ginzburg-Landau equation framework.
- Analyzing the system's response to spatially localized initial conditions.
- Systematically varying feedback control strength and delay parameters.
Main Results:
- Feedback control successfully stabilizes initially localized solutions.
- Increasing delay induces oscillatory instability in the localized solutions.
- For sufficient control strength and delay, localized oscillating pulses are formed.
- Further increases in delay lead to solution blow-up.
Conclusions:
- Delayed feedback control can stabilize otherwise unstable localized patterns.
- The interplay between control delay and strength dictates the emergent dynamics, from stable patterns to oscillations and blow-up.
- This study provides insights into controlling complex spatio-temporal phenomena.
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