Hopf bifurcation in quasilinear reaction-diffusion systems

1991 ◽  
pp. 53-63 ◽  
Author(s):  
Herbert Amann
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

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 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.

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