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December 2005 Relative Poincaré-Hopf bifurcation and Galloping Instability of Traveling Waves
Benjamin Texier, Kevin Zumbrun
Methods Appl. Anal. 12(4): 349-380 (December 2005).


For a class of reaction–convection–diffusion systems studied by Sattinger, notably including Majda’s model of reacting flow, we rigorously characterize the transition from stability to time-periodic “galloping” instability of traveling-wave solutions as a relative Poincaré–Hopf bifurcation arising in ODE with a group invariance– in this case, translational symmetry. More generally, we show how to construct a finite-dimensional center manifold for a second-order parabolic evolution equation inheriting an underlying group invariance of the PDE, by working with a canonical integro-differential equation induced on the quotient space. This reduces the questions of existence and stability of bounded solutions of the PDE to existence and stability of solutions of the reduced, finite-dimensional ODE on the center manifold, which may then be studied by more standard, finite-dimensional bifurcation techniques.


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Benjamin Texier. Kevin Zumbrun. "Relative Poincaré-Hopf bifurcation and Galloping Instability of Traveling Waves." Methods Appl. Anal. 12 (4) 349 - 380, December 2005.


Published: December 2005
First available in Project Euclid: 5 April 2007

zbMATH: 1043.90032
MathSciNet: MR2258314

Primary: 34C20 , 35B10 , 35B32 , 76L05

Keywords: center manifold reduction , detonation waves , galloping instability , Poincaré-Hopf bifurcation , Stability of traveling waves , weighted norms

Rights: Copyright © 2005 International Press of Boston


Vol.12 • No. 4 • December 2005
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