  Motivated by physical and numerical observations of time oscillatory
``galloping'', ``spinning'', and ``cellular'' instabilities of detonation
waves, we study Poincar\'e--Hopf bifurcation of traveling-wave solutions of
viscous conservation laws. The main difficulty is the absence of a spectral gap
between oscillatory modes and essential spectrum, preventing standard reduction
to a finite-dimensional center manifold. We overcome this by direct
Lyapunov--Schmidt reduction, using detailed pointwise bounds on the linearized
solution operator to carry out a nonstandard implicit function construction in
the absence of a spectral gap. The key computation is a space-time stability
estimate on the transverse linearized solution operator reminiscent of Duhamel
estimates carried out on the full solution operator in the study of nonlinear
stability of spectrally stable traveling waves.
