Nonlinear dynamical systems are rather common in engineering. This class of problems is usually solved by numerical integration or through the use of ap- proximate analytical methods (perturbation methods) or semi-analytical meth- ods such as the harmonic balance method. The numerical integration is a slow and cumbersome process in parametric analyses. The other methods are usu- ally extremely fast but they are less precise and their application to problems involving certain types of non-linearity, such as fractional power non-linearities, are difficult or even impossible. In this work two alternative methodologies for the analysis of non-linear dynamical systems, based on Taylor series expan- sions, are proposed. In the first method, the solution of the initial value prob- lem is obtained by expanding the response in Taylor series and the symmetries of the response in phase space are used to obtain the frequency-amplitude rela- tion or the fixed points of the steady-state response. In the second method the response is written as a Fourier series and the modal amplitudes are obtained using the same methodology used in the previous method for the determina- tion of the coefficients of the Taylor expansion. The symmetries of the response are implicit in the Fourier series, and supplementary equations are proposed for the determination of the frequency-amplitude relation and the fixed points of the response. Comparisons with other existing methods show that the two proposed methods are precise and can be easily applied to the analysis of sev- eral dynamical systems. The main advantages of the proposed methods are that they can be applied to several types of non-linearities and that analytic expression for the Fourier coefficients can be obtained by the solution of a system of linear algebraic equations.