A method of initialvalue transformation was presented to obtain the approximate analytic periods of a class of nonlinear oscillators. The periodic solutions 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 a frequency, amplitudes as the independent variables using RitzGalerkin’s method. But the set of equations is incomplete, and the key is to consider initial value transformation. After adding supplementary equations, a set of nonlinear algebraic equations with angular frequencies, amplitudes as the independent variables was constituted completely. For examples, six asymmetric periodic solutions bifurcating about a nonlinear differential equation arising in general relativity were solved by using the method of initialvalue transform. Amplitudefrequency curves and central offsetfrequency curves of the asymmetrically vibration systems were derived. In addition, the drift phenomenon of natural angular frequency was discovered.