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Quantitative and Temporal Control of Oxygen Microenvironment at the Single Islet Level

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Published on: November 17, 2013

Time-delayed feedback control of breathing localized structures in a three-component reaction-diffusion system.

Svetlana V Gurevich¹

¹Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany gurevics@uni-muenster.de.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences

|September 24, 2014

Summary

Controlling breathing localized structures in reaction-diffusion systems is possible by adjusting time-delayed feedback. This research demonstrates how delay time and feedback strength can stabilize structures or induce oscillations.

Keywords:

bifurcation analysis localized structures reaction–diffusion systems time-delayed feedback

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Area of Science:

Complex Systems
Nonlinear Dynamics
Chemical Kinetics

Background:

Reaction-diffusion systems exhibit complex spatio-temporal patterns.
Localized structures in these systems can display dynamic behaviors like breathing.
Time-delayed feedback is a common control mechanism in dynamical systems.

Purpose of the Study:

To investigate the dynamics of a breathing localized structure in a three-component reaction-diffusion system with time-delayed feedback.
To determine how delay time and feedback strength influence the stability and oscillation of the localized structure.
To derive a control mechanism for manipulating the behavior of these structures.

Main Methods:

Bifurcation analysis of the reaction-diffusion system.
Derivation of an order parameter equation near the Andronov-Hopf bifurcation.
Mathematical analysis to derive stabilization domains and oscillation dependencies.

Main Results:

Variation of delay time and feedback strength can stabilize breathing localized structures.
Delay-induced periodic or quasi-periodic oscillations of localized structures were observed.
An order parameter equation accurately describes dynamics near bifurcations.
Explicit derivation of stabilization domain boundaries and oscillation radius dependence on delay parameters.

Conclusions:

Time-delayed feedback offers a robust and straightforward mechanism to control breathing localized structures.
The derived order parameter equation provides analytical insight into stabilization and oscillation phenomena.
This work contributes to understanding and controlling complex dynamics in reaction-diffusion systems.