Hopf Bifurcation in Spatially Extended Reaction—Diffusion Systems

1998 ◽  
Vol 8 (1) ◽  
pp. 17-41 ◽  
Author(s):  
G. Schneider
Keyword(s):  
Hopf Bifurcation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatially Extended

A New Mechanical Algorithm for Calculating the Amplitude Equation of the Reaction-Diffusion Systems

2011 ◽  
pp. 205-213
Author(s):  
Houye Liu ◽  
Weiming Wang
Keyword(s):  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Normal Form ◽  
Reaction Diffusion ◽  
Amplitude Equation ◽  
Diffusion Reaction ◽  
Future Studies ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Form Approach

Amplitude equation may be used to study pattern formatio. In this chapter, we establish a new mechanical algorithm AE_Hopf for calculating the amplitude equation near Hopf bifurcation based on the method of normal form approach in Maple. The normal form approach needs a large number of variables and intricate calculations. As a result, deriving the amplitude equation from diffusion-reaction is a difficult task. Making use of our mechanical algorithm, we derived the amplitude equations from several biology and physics models. The results indicate that the algorithm is easy to apply and effective. This algorithm may be useful for learning the dynamics of pattern formation of reaction-diffusion systems in future studies.

scholarly journals Incorporating domain growth into hybrid methods for reaction–diffusion systems

2021 ◽  
Vol 18 (177) ◽  
Author(s):  
Cameron A. Smith ◽  
Christian A. Yates
Keyword(s):  
Hybrid Methods ◽  
Spatial Scales ◽  
Reaction Diffusion ◽  
Domain Growth ◽  
Reaction Diffusion Systems ◽  
Multi Scale ◽  
Diffusion Systems ◽  
Spatially Extended ◽  
Physical Phenomena ◽  
Migration Of Cells

Reaction–diffusion mechanisms are a robust paradigm that can be used to represent many biological and physical phenomena over multiple spatial scales. Applications include intracellular dynamics, the migration of cells and the patterns formed by vegetation in semi-arid landscapes. Moreover, domain growth is an important process for embryonic growth and wound healing. There are many numerical modelling frameworks capable of simulating such systems on growing domains; however, each of these may be well suited to different spatial scales and particle numbers. Recently, spatially extended hybrid methods on static domains have been produced to bridge the gap between these different modelling paradigms in order to represent multi-scale phenomena. However, such methods have not been developed with domain growth in mind. In this paper, we develop three hybrid methods on growing domains, extending three of the prominent static-domain hybrid methods. We also provide detailed algorithms to allow others to employ them. We demonstrate that the methods are able to accurately model three representative reaction–diffusion systems accurately and without bias.

scholarly journals Hopf bifurcation of travelling pulses in some bistable reaction-diffusion systems

2000 ◽  
Vol 7 (1) ◽  
pp. 165-194 ◽  
Author(s):  
Tsutomu Ikeda ◽  
Hideo Ikeda ◽  
Masayasu Mimura
Keyword(s):  
Hopf Bifurcation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems

scholarly journals Interface oscillations in reaction-diffusion systems above the Hopf bifurcation

2012 ◽  
Vol 17 (7) ◽  
pp. 2523-2543 ◽  
Author(s):  
Rebecca McKay ◽  
◽  
Theodore Kolokolnikov ◽  
Paul Muir ◽  
◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems

A New Mechanical Algorithm for Calculating the Amplitude Equation of the Reaction-Diffusion Systems

2012 ◽  
Vol 3 (4) ◽  
pp. 21-28
Author(s):  
Houye Liu ◽  
Weiming Wang
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Reaction Diffusion ◽  
Amplitude Equation ◽  
Diffusion Reaction ◽  
Amplitude Equations ◽  
Future Studies ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Form Approach

Amplitude equation may be used to study pattern formatio. In this article, the authors establish a new mechanical algorithm AE_Hopf for calculating the amplitude equation near Hopf bifurcation based on the method of normal form approach in Maple. The normal form approach needs a large number of variables and intricate calculations. As a result, deriving the amplitude equation from diffusion-reaction is a difficult task. Making use of our mechanical algorithm, we derived the amplitude equations from several biology and physics models. The results indicate that the algorithm is easy to apply and effective. This algorithm may be useful for learning the dynamics of pattern formation of reaction-diffusion systems in future studies.

ON TURING-HOPF INSTABILITIES IN REACTION-DIFFUSION SYSTEMS

2008 ◽  
Vol 03 (01n02) ◽  
pp. 257-274 ◽  
Author(s):  
MARIANO RODRÍGUEZ RICARD
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Limit Cycles ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Hopf Bifurcations ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Turing Instabilities ◽  
Spatially Homogeneous

We examine the appearance of Turing instabilities of spatially homogeneous periodic solutions in reaction-diffusion equations when such periodic solutions are consequence of Hopf bifurcations. First, we asymptotically develop limit cycle solutions associated to the appearance of Hopf bifurcations in reaction systems. Particularly, we will show conditions to the appearance of multiple limit cycles after Hopf bifurcation. Then, we propose expansions to normal modes associated with Turing instabilities from spatially homogeneous periodic solutions associated to limit cycles which appear as a consequence of a Hopf bifurcation. Finally, we discuss examples of reaction-diffusion systems arising in biology and chemistry in which can be observed spatial and time-periodic patterning.

CHUA'S CIRCUITS SYNCHRONIZATION WITH DIFFUSIVE COUPLING: NEW RESULTS

2009 ◽  
Vol 19 (09) ◽  
pp. 3103-3107 ◽  
Author(s):  
ARTURO BUSCARINO ◽  
LUIGI FORTUNA ◽  
MATTIA FRASCA ◽  
GREGORIO SCIUTO
Keyword(s):  
Reaction Diffusion ◽  
Chua’S Circuit ◽  
Experimental Realization ◽  
Chua's Circuit ◽  
Reaction Diffusion Systems ◽  
Diffusive Coupling ◽  
Nonlinear Network ◽  
Diffusion Systems ◽  
Spatially Extended ◽  
Points Of View

In this communication, synchronization of two diffusively coupled Chua's circuit is studied from the analytical and experimental points of view. The conditions under which complete synchronization is ensured are derived by applying a strategy based on the Master Stability Function. The experimental realization, that makes use of the State-Controlled Cellular Nonlinear Network based implementation of Chua's circuit, shows the synchronized behavior of two circuits coupled with a passive resistor as diffusion coefficient. The results obtained indicates diffusive coupling as a mutually reduced order state observer, in the sense that one circuit observes the other and vice versa. Moreover, the concept of synchronization by using passive elements can be extended to spatially extended reaction–diffusion systems.

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