Existence of periodic solutions for semilinear reaction diffusion systems

2004 ◽  
Vol 59 (6) ◽  
pp. 931-949 ◽  
Author(s):  
Norimichi Hirano ◽  
Sławomir Rybicki
Keyword(s):  
Periodic Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Existence Of Periodic Solutions

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.

scholarly journals Periodic traveling fronts for partially degenerate reaction-diffusion systems with bistable and time-periodic nonlinearity

2019 ◽  
Vol 9 (1) ◽  
pp. 923-957
Author(s):  
Shi-Liang Wu ◽  
Cheng-Hsiung Hsu
Keyword(s):  
Periodic Solutions ◽  
Wave Speed ◽  
Reaction Diffusion ◽  
Wave Speeds ◽  
Reaction Diffusion Systems ◽  
Liapunov Stability ◽  
Traveling Fronts ◽  
Diffusion Systems ◽  
Stable Periodic ◽  
Time Periodic

Abstract This paper is concerned with the periodic traveling fronts for partially degenerate reaction-diffusion systems with bistable and time-periodic nonlinearity. We first determine the signs of wave speeds for two monostable periodic traveling fronts of the system. Then, we prove the existence of periodic traveling fronts connecting two stable periodic solutions. An estimate of the wave speed is also obtained. Further, we prove the monotonicity, uniqueness (up to a translation), Liapunov stability and exponentially asymptotical stability of the smooth bistable periodic traveling fronts.

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