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Hopf bifurcation for semilinear dissipative hyperbolic systems. (English) Zbl 1310.35030

Hopf bifurcation is considered for the time-periodic solutions of semilinear hyperbolic systems of the type \[ \partial_t u_j + a_j(x,\lambda)\partial_x u_j + b_j(x,\lambda,u) = 0,\quad x \in (0,1),\; j = 1,\dots,n, \] depending upon a real parameter \(\lambda\), when \[ b_j(x,\lambda,0) = 0 \] for all \(x \in [0,1]\) and \(\lambda \in \mathbb R\). The space boundary conditions are of reflection type.
The approach uses Lyapunov-Schmidt method, implicit function arguments and the fiber contraction principle. An explicit formula determines the bifurcation direction. The result is compared with related works.

MSC:

35B32 Bifurcations in context of PDEs
47J15 Abstract bifurcation theory involving nonlinear operators
35L50 Initial-boundary value problems for first-order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
35B10 Periodic solutions to PDEs
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