In the present paper eigenvalue problems for non-selfadjoint linear differential operators smoothly dependent on a vector of real parameters are considered. Bifurcation of eigenvalues along smooth curves in the parameter space is studied. The case of m ultipleeigen value with Keldysh chain of arbitrary length is considered. Explicit expressions describing bifurcation of eigenvalues are found. The obtained formulae use eigenfunctions and associated functions of the adjoint eigenvalue problems as well as the derivatives of the differential operator taken at the initial point of the parameter space. These results are important for the stabilit y theory, sensitivit y analysis and structural optimization. As a mechanical application the extended Beck's problem of stabilit y of an elastic column under action of potential force and tangential folio w er force is considered and discussed in detail.
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