Weakly Nonlinear Oscillations with Analytic Forcing

Weakly nonlinear oscillations exhibit a wide range of phenomena not seen in simple linear oscillations. This paper considers weakly nonlinear oscillations with an analytic forcing term, and attempts to understand various quantitative solution methods for this problem. We give a quantitative demonstration of the failure of regular perturbation theory, and use this failure to motivate investigation into two-timing and averaging theory. We see that both two-timing and averaging theory give initially identical approximations of a solution, in fact giving excellent approximations even when the nonlinear effects cease to be weak. Finally, we see that the results given by two-timing and averaging theory are physically valid only when the exact solution is bounded.