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We show that transition to longitudinal instability of strong detonationsolutions of reactive compressible Navier--Stokes equations is genericallyassociated with Hopf bifurcation to nearby time-periodic "galloping", or"pulsating", solutions, in agreement with physical and numerical observation.In the process, we determine readily numerically verifiable stability andbifurcation conditions in terms of an associated Evans function, and obtain thefirst complete nonlinear stability result for strong detonations of thereacting Navier--Stokes equations, in the limit as amplitude (hence also heatrelease) goes to zero. The analysis is by pointwise semigroup techniquesintroduced by the authors and collaborators in previous works.

Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions