Hopf bifurcation for semilinear dissipative hyperbolic systems

We consider boundary value problems for semilinear hyperbolic systems of the type∂tuj+aj(x,λ)∂xuj+bj(x,λ,u)=0,x∈(0,1),j=1,…,n, with smooth coefficient functions ajaj and bjbj such that bj(x,λ,0)=0bj(x,λ,0)=0 for all x∈[0,1]x∈[0,1], λ∈Rλ∈R, and j=1,…,nj=1,…,n. We state conditions for Hopf bifurcation, i.e., for existence, local uniqueness (up to phase shifts), smoothness and smooth dependence on λ of time-periodic solutions bifurcating from the zero stationary solution. Furthermore, we derive a formula which determines the bifurcation direction.The proof is done by means of a Liapunov–Schmidt reduction procedure. For this purpose, Fredholm properties of the linearized system and implicit function theorem techniques are used.