This talk is about the efficient detection of Hopf bifurcation points and computation of periodic orbits in large-scale dynamical systems that might arise after spatial discretization of PDEs. Several different approaches are discussed: 1) methods based on locating high-order singularities (``organising centres"), 2) methods based on the direct calculation of dangerous eigenvalues, 3) methods based on counting ideas from complex analysis, and 4) Reduced basis methods.

Examples will be given from a variety of physical problems, including large systems arising from mixed finite-element discretizations of the Navier-Stokes equations.

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