Frequency of free magnetic vortex precession in circular soft ferromagnetic nano-cylinders (magnetic dots) of various sizes is an important parameter, used in design of spintronic devices (such as spin-torque microwave nano-oscillators) and characterization of magnetic nanostructures. Here, using a recently developed collective-variable approach to non-linear dynamics of magnetic textures in planar nano-magnets, this frequency and its amplitude-dependent shift are computed analytically and

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... ed for the full range of cylinder geometries. The frequency shift is positive in large planar dots, but becomes negative in smaller and more elongated ones. At certain dot dimensions a zero frequency shift is realized, which can be important for enhancing frequency stability of magnetic nano-oscillators.