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We present a self-contained formalism for analyzing scale invariantdifferential equations. We first cast the scale invariant model into itsequidimensional and autonomous forms, find its fixed points, and then obtainpower-law background solutions. After linearizing about these fixed points, wefind a second linearized solution, which provides a distinct collection ofpower laws characterizing the deviations from the fixed point. We prove thatgenerically there will be a region surrounding the fixed point in which thecomplete general solution can be represented as a generalized Frobenius-likepower series with exponents that are integer multiples of the exponents arisingin the linearized problem. This Frobenius-like series can be viewed as avariant of Liapunov's expansion theorem. As specific examples we apply theseideas to Newtonian and relativistic isothermal stars and demonstrate (bothnumerically and analytically) that the solution exhibits oscillatory power-lawbehaviour as the star approaches the point of collapse. These series solutionsextend classical results. (Lane, Emden, and Chandrasekhar in the Newtoniancase; Harrison, Thorne, Wakano, and Wheeler in the relativistic case.) We alsoindicate how to extend these ideas to situations where fixed points may notexist -- either due to ``monotone'' flow or due to the presence of limitcycles. Monotone flow generically leads to logarithmic deviations from scaling,while limit cycles generally lead to discrete self-similar solutions.Comment: 35 pages; IJMPA style fil

Power Laws, Scale Invariance and the Generalized Frobenius Series: Applications to Newtonian and TOV Stars Near Criticality