Methods and Applications of Analysis
Volume 12 (2005)
Relative Poincaré-Hopf bifurcation and Galloping Instability of Traveling Waves
Pages: 349 – 380
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.
Stability of traveling waves; detonation waves; weighted norms; Poincaré-Hopf bifurcation; center manifold reduction; galloping instability
2010 Mathematics Subject Classification
34C20, 35B10, 35B32, 76L05