American Institute of Mathematical Sciences

Advanced
2007, 2007(Special): 784-793. doi: 10.3934/proc.2007.2007.784

Degenerate Hopf instability in oscillatory reaction-diffusion equations

Toshi Ogawa 1,

1. 

Graduate school of Engineering Science, Osaka University, Toyonaka, 560-8531, Japan

Received  September 2006 Revised  May 2007 Published  September 2007

Degenerate critical points on a 2-compnent reaction diffusion system with a global inhibition (GI) are studied. They can be an organizing center for a variety of spatially non-uniform oscillations. The Hopf-bifurcation point for the 0-mode oscillation (spatially uniform oscillation) is controlled by the strength of GI so that the eairlier destabilization of the higher mode oscillations can happen.
Keywords: normal form, Degenerate Hopf bifurcation, rotating wave, standing wave..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Toshi Ogawa. Degenerate Hopf instability in oscillatory reaction-diffusion equations. Conference Publications, 2007, 2007 (Special) : 784-793. doi: 10.3934/proc.2007.2007.784
[1]

Kazuyuki Yagasaki. Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2621-2634. doi: 10.3934/dcdsb.2021151
[2]

Gabriela Jaramillo. Rotating spirals in oscillatory media with nonlocal interactions and their normal form. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022085
[3]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208
[4]

Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3271-3284. doi: 10.3934/dcdss.2020129
[5]

Jong-Shenq Guo, Hirokazu Ninomiya, Chin-Chin Wu. Existence of a rotating wave pattern in a disk for a wave front interaction model. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1049-1063. doi: 10.3934/cpaa.2013.12.1049
[6]

Hyungjin Huh. A special form of solution to half-wave equations. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021056
[7]

Jaeyoung Byeon, Ohsang Kwon, Yoshihito Oshita. Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations. Communications on Pure and Applied Analysis, 2015, 14 (3) : 825-842. doi: 10.3934/cpaa.2015.14.825
[8]

Guido Schneider, Matthias Winter. The amplitude system for a Simultaneous short-wave Turing and long-wave Hopf instability. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021119
[9]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[10]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[11]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[12]

Renata Bunoiu, Radu Precup, Csaba Varga. Multiple positive standing wave solutions for schrödinger equations with oscillating state-dependent potentials. Communications on Pure and Applied Analysis, 2017, 16 (3) : 953-972. doi: 10.3934/cpaa.2017046
[13]

Hiroaki Kikuchi. Remarks on the orbital instability of standing waves for the wave-Schrödinger system in higher dimensions. Communications on Pure and Applied Analysis, 2010, 9 (2) : 351-364. doi: 10.3934/cpaa.2010.9.351
[14]

Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2411-2445. doi: 10.3934/dcds.2020370
[15]

Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100
[16]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete and Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363
[17]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643
[18]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[19]

Xiaojie Hou, Yi Li, Kenneth R. Meyer. Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 265-290. doi: 10.3934/dcds.2010.26.265
[20]

Faustino Sánchez-Garduño, Philip K. Maini, Judith Pérez-Velázquez. A non-linear degenerate equation for direct aggregation and traveling wave dynamics. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 455-487. doi: 10.3934/dcdsb.2010.13.455

 Impact Factor: 

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy