The periodic solutions of a class of nonlinear oscillators can be expressed in the forms of basic harmonics and bifurcate harmonics. Thus, an oscillation system which is described as a second order ordinary differential equation, can be expressed as a set of non-linear algebraic equations with frequency, central offset and amplitudes as the independent variables by using Ritz-Galerkin method. But the set of equations is incomplete. The key is that considering initial -value transformation, supplementary equations. were added and a set of non-linear algebraic equations with angular frequencies and amplitudes as the independent variables was constituted completely. As examples, six asymmetric periodic solutions bifurcating about a nonlinear differential equation arising in general relativity were solved by using the method of initial-value transformation. Amplitude-frequency curves and central offset-frequency curves of the asymmetrical vibration systems were derived. In addition, the drift phenomenon of natural angular frequency was discovered.