Analytical approach to localized structures in a simple reaction-diffusion system

2004 ◽  
Vol 69 (2) ◽  
Author(s):  
Orazio Descalzi ◽  
Yumino Hayase ◽  
Helmut R. Brand
Keyword(s):  
Analytical Approach ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Simple Reaction ◽  
Localized Structures

Pattern formation arising from wave instability in a simple reaction‐diffusion system

1995 ◽  
Vol 103 (23) ◽  
pp. 10306-10314 ◽  
Author(s):  
Anatol M. Zhabotinsky ◽  
Milos Dolnik ◽  
Irving R. Epstein
Keyword(s):  
Pattern Formation ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Wave Instability ◽  
Simple Reaction

scholarly journals Oscillatory Turing Patterns in a Simple Reaction-Diffusion System

2007 ◽  
Vol 50 (91) ◽  
pp. 234 ◽  
Author(s):  
Ruey-Tarng Ruey-Tarng ◽  
Sy-Sang Sy-Sang ◽  
Philip K.
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Turing Patterns ◽  
Simple Reaction

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

2014 ◽  
Vol 372 (2027) ◽  
pp. 20140014 ◽  
Author(s):  
Svetlana V. Gurevich
Keyword(s):  
Reaction Diffusion ◽  
Delayed Feedback ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Delayed Feedback Control ◽  
Straightforward Manner ◽  
Localized Structures ◽  
Feedback Strength ◽  
Three Component Reaction ◽  
Time Delayed Feedback

The dynamics of a single breathing localized structure in a three-component reaction–diffusion system subjected to time-delayed feedback is investigated. It is shown that variation of the delay time and the feedback strength can lead either to stabilization of the breathing or to delay-induced periodic or quasi-periodic oscillations of the localized structure. A bifurcation analysis of the system in question is provided and an order parameter equation is derived that describes the dynamics of the localized structure in the vicinity of the Andronov–Hopf bifurcation. With the aid of this equation, the boundaries of the stabilization domains as well as the dependence of the oscillation radius on delay parameters can be explicitly derived, providing a robust mechanism to control the behaviour of the breathing localized structure in a straightforward manner.

Stabilization of Dominant Structures in an Ionic Reaction-Diffusion System

1998 ◽  
Vol 63 (6) ◽  
pp. 761-769 ◽  
Author(s):  
Roland Krämer ◽  
Arno F. Münster
Keyword(s):  
Reaction Diffusion ◽  
Dominant Mode ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Local Perturbation ◽  
Dimensional System ◽  
One Dimensional ◽  
Brusselator Model ◽  
The One ◽  
Dominant Structure

We describe a method of stabilizing the dominant structure in a chaotic reaction-diffusion system, where the underlying nonlinear dynamics needs not to be known. The dominant mode is identified by the Karhunen-Loeve decomposition, also known as orthogonal decomposition. Using a ionic version of the Brusselator model in a spatially one-dimensional system, our control strategy is based on perturbations derived from the amplitude function of the dominant spatial mode. The perturbation is used in two different ways: A global perturbation is realized by forcing an electric current through the one-dimensional system, whereas the local perturbation is performed by modulating concentrations of the autocatalyst at the boundaries. Only the global method enhances the contribution of the dominant mode to the total fluctuation energy. On the other hand, the local method leads to simple bulk oscillation of the entire system.

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