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This work presents an analytical perturbation method to define and study the dynamics of frozen orbits under the perturbation effects produced by the oblatness of the main celestial body. This is done using a perturbation method purely based on osculating elements. This allows to characterize, define, and study the three existing families of frozen orbits in closed-form: the two families of frozen orbits close to the critical inclination, and the family of frozen orbits that appears at low values of eccentricity. To that end, this work includes the first and second order approximate solutions of the proposed perturbation method, including their applications to define frozen orbits, repeating ground-track orbits, and sun-synchronous orbits. Examples of application are also presented to show the expected error performance of the proposed approach.