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Controlling pulse stability in singularly perturbed reaction-diffusion systems.

F Veerman¹, I Schneider²

¹Leiden University, Mathematisch Instituut, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands.

Chaos (Woodbury, N.Y.)

|December 7, 2023

Summary

This summary is machine-generated.

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Pyragas control stabilizes unstable localized structures in reaction-diffusion systems. This feedback method enhances the stability of singular pulses across a wide parameter range.

Area of Science:

Nonlinear Dynamics
Chemical Kinetics
Mathematical Physics

Background:

Reaction-diffusion systems exhibit complex spatio-temporal dynamics, including localized coherent structures.
Singularly perturbed systems present challenges in stability analysis due to fast and slow scales.
Localized structures like pulses are crucial in various phenomena but can be unstable.

Purpose of the Study:

To investigate the efficacy of Pyragas control for stabilizing stationary, localized coherent structures.
To apply noninvasive Pyragas-like proportional feedback control to a specific two-component reaction-diffusion system.
To determine the parameter space where control can stabilize an otherwise unstable singular pulse solution.

Main Methods:

Utilizing Pyragas-like proportional feedback control, a noninvasive technique.

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Analyzing a general class of two-component, singularly perturbed reaction-diffusion systems.

Investigating the stability of a singular pulse solution under control action.

Main Results:

Demonstrated successful stabilization of a singular pulse solution.
Identified a significant region of parameter space where control is effective.
Showcased the ability to adjust control parameters to achieve stability.

Conclusions:

Pyragas control is a viable method for enhancing the stability of localized structures in reaction-diffusion systems.
The control strategy is effective in stabilizing otherwise unstable singular pulses.
This approach offers a pathway to control complex dynamics in nonlinear systems.