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Feedback control of subcritical oscillatory instabilities
A A Golovin1, A A Nepomnyashchy
1Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA.
Summary
Feedback control using a complex Ginzburg-Landau equation suppresses instabilities, forming localized pulses. Different pulse dynamics emerge based on parameters and feedback delay, impacting stability and amplitude.
Area of Science:
- Nonlinear dynamics
- Complex systems
- Pattern formation
Background:
- Subcritical oscillatory instabilities can lead to pattern formation.
- Complex Ginzburg-Landau equation models nonlinear dynamics near instability thresholds.
- Feedback control offers a method to manage these instabilities.
Purpose of the Study:
- Investigate feedback control of oscillatory instability.
- Analyze the formation of localized pulses (oscillons) via control.
- Explore the impact of control parameters and feedback delay on dynamics.
Main Methods:
- Utilized a globally-controlled complex Ginzburg-Landau equation.
- Implemented feedback loop linking linear growth rate and pattern amplitude.
- Studied one-dimensional and two-dimensional systems.
- Analyzed the effect of feedback delay.
Main Results:
- Feedback control successfully suppressed blow-up and formed localized pulses.
- Observed diverse pulse dynamics: stationary, coexisting, competing, chaotic, and synchronized.
- Two-dimensional systems showed similar dynamic behaviors.
- Increased feedback delay induced pulse instability and amplitude oscillations.
Conclusions:
- Feedback control is effective for stabilizing oscillatory instabilities and generating localized structures.
- System parameters and feedback delay critically influence pulse dynamics and stability.
- This control strategy offers a pathway to engineer complex spatiotemporal patterns.