Interaction of dissipative solitons: particle-like behaviour of localized structures in a three-component reaction-diffusion system

2002 ◽  
Vol 161 (1-2) ◽  
pp. 45-66 ◽  
Author(s):  
M. Bode ◽  
A.W. Liehr ◽  
C.P. Schenk ◽  
H.-G. Purwins
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Localized Structures ◽  
Dissipative Solitons ◽  
Three Component 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.

AUTOWAVES AND SOLITONS IN A THREE-COMPONENT REACTION–DIFFUSION SYSTEM

2002 ◽  
Vol 12 (11) ◽  
pp. 2421-2434 ◽  
Author(s):  
V. I. NEKORKIN ◽  
V. B. KAZANTSEV
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Order System ◽  
Qualitative Explanation ◽  
Nonlinear Odes ◽  
Traveling Pulses ◽  
Dissipative Solitons ◽  
Three Component Reaction ◽  
Velocity Spectra

We study the propagation of nonlinear waves in a three-component reaction–diffusion system. The problem of the existence of the stationary pulse-like solutions is reduced to the analysis of homoclinic trajectories of a fourth-order system of nonlinear ODEs. We have obtained the parameter set corresponding to the homoclinic bifurcations that defines the velocity spectra of the traveling pulses. We have shown that the pulses behave like autowaves annihilating in head-on collision and like dissipative solitons crossing each other, reflecting at boundaries. We have provided a qualitative explanation for such a behavior.

Butterfly Catastrophe for Fronts in a Three-Component Reaction–Diffusion System

2014 ◽  
Vol 25 (1) ◽  
pp. 87-129 ◽  
Author(s):  
Martina Chirilus-Bruckner ◽  
Arjen Doelman ◽  
Peter van Heijster ◽  
Jens D. M. Rademacher
Keyword(s):  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Three Component Reaction
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