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Feedback control of subcritical Turing instability with zero mode.

A A Golovin1, Y Kanevsky, A A Nepomnyashchy

  • 1Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
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Global feedback control stabilizes systems with Turing instability using pattern amplitude measurements. This leads to stable localized patterns or traveling waves, crucial for pattern formation studies.

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Area of Science:

  • Nonlinear dynamics
  • Pattern formation
  • Control theory

Background:

  • Systems with subcritical instabilities can exhibit complex behaviors.
  • Turing instability (short-wave instability) is a key mechanism for pattern formation.
  • Zero modes can significantly influence system dynamics.

Purpose of the Study:

  • Investigate global feedback control for systems with Turing instability and a zero mode.
  • Determine if feedback control can stabilize these systems.
  • Analyze the resulting stationary and traveling states.

Main Methods:

  • Utilized a Ginzburg-Landau equation coupled with a zero mode equation.
  • Employed analytical and numerical simulation techniques.
  • Implemented feedback control based on maximum pattern amplitude.

Main Results:

  • Feedback control successfully stabilized the system.
  • Localized unipulse stationary states were formed.
  • Traveling solitary waves emerged due to an instability of stationary states.

Conclusions:

  • Global feedback control is effective for stabilizing systems with Turing instability and zero modes.
  • The transition to traveling waves is governed by the coupling parameter and zero mode damping.
  • This control strategy enables the formation of predictable localized structures.