TY - GEN A1 - Kmit, Irina A1 - Recke, Lutz T1 - Periodic Solutions to Dissipative Hyperbolic Systems. II: Hopf Bifurcation for Semilinear Problems N2 - We consider boundary value problems for semilinear hyperbolic systems of the type $$ \partial_tu_j  + a_j(x,\la)\partial_xu_j + b_j(x,\la,u) = 0, \; x\in(0,1), \;j=1,\dots,n $$ with smooth coefficient functions $a_j$ and $b_j$ such that $b_j(x,\la,0) = 0$ for all $x \in [0,1]$, $\la \in \R$, and $j=1,\ldots,n$. We state conditions for Hopf bifurcation, i.e., for existence, local uniqueness (up to phase shifts), smoothness and smooth dependence on $\la$ 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. There are at least two distinguishing features of Hopf bifurcation theorems for hyperbolic PDEs in comparison with those for parabolic PDEs or for ODEs: First, the question if a non-degenerate time-periodic solution depends smoothly on the system parameters is much more delicate. And second, a sufficient amount of dissipativity is needed in the system, and a priori it is not clear how to verify this in terms of the data of the PDEs and of the boundary conditions. KW - first-order hyperbolic systems KW - time-periodic solutions KW - reflection boundary conditions KW - Liapunov-Schmidt procedure KW - implicit function theorem KW - fiber contraction principle Y1 - 2013 UR - https://opus4.kobv.de/opus4-matheon/frontdoor/index/index/docId/1198 UR - https://nbn-resolving.org/urn:nbn:de:0296-matheon-11981 ER -