Mathematical Analysis of a Singular, Nonlinear, Periodically Driven Oscillator

We investigate the possible solutions of the second-order differential equation \begin{equation} m\overset{\cdot \cdot }{x}+\overset{\cdot }{x}+x^{3}=\sin t, \tag{*} \end{equation}% for the limiting case where $m=0$. Applying the method of harmonic balance [1], we determine both first- and second-order approximations to the periodic solution. We also show, using the qualitative theory of differential equations [2], that this periodic solution is an attractor, i.e., regardless of the initial condition, $x_{0}=x(0),$ the solution eventually becomes arbitrarily close to this periodic solution. This work extends the results of Elias [3]. \bigskip [1] \ Ronald E. Mickens., \textit{Oscillations in Planar Dynamic Systems} (World Scientific, Singapore, 1996); see Chapter 4. [2] \ See ref. [1], Appendix I. [3] \ U. Elias, \textit{Qualitative analysis of a differential equation of Abel}, MAA Monthly (February 2008), pps. 147-149.