Analysis of a weakly non-linear autonomous oscillator by means of the field method

In this paper the field method is extended to the study of oscillatory systems with two degrees of freedom and weak quadratic non-linearity. The basic field method concept is combined with the technique of multiple time scales and the solution for both non-resonant case and the case out of the first resonance are found. The qualitative analysis of behavior in the resonant area is done by determining the values of the “adelphic” integral.