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Eigenvalue veering is studied in the context of two simple oscillators coupled by a (presumed weak) spring, variants of which have been considered by several authors. The concept of a center of veering is introduced, leading to a coordinate translation; a subsequent coordinate rotation, dependent on the degree of asymmetry of the system, reduces the frequency equation to a standard north-south opening hyperbola. Thus veering occurs even when coupling is strong, and may be characterized by these coordinate transformations and geometric features of the hyperbola, rather than eigenvalue and eigenvector derivatives.
N.G. Stephen (Thu,) studied this question.