Methods and Applications of Analysis

Relative Poincaré-Hopf bifurcation and Galloping Instability of Traveling Waves

Benjamin Texier and Kevin Zumbrun

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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.

Article information

Methods Appl. Anal., Volume 12, Number 4 (2005), 349-380.

First available in Project Euclid: 5 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 76L05: Shock waves and blast waves [See also 35L67] 35B32: Bifurcation [See also 37Gxx, 37K50] 35B10: Periodic solutions 34C20: Transformation and reduction of equations and systems, normal forms

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


Texier, Benjamin; Zumbrun, Kevin. Relative Poincaré-Hopf bifurcation and Galloping Instability of Traveling Waves. Methods Appl. Anal. 12 (2005), no. 4, 349--380.

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