Approximate Solutions to d$^{2}$x/dt$^{2}$ + [1+( dx/dt)$^{2}$]x = 0 Using a Polar Representation

It can be shown that the following nonlinear differential equation \begin{center} d$^{2}$x/dt$^{2}$ + [1+( dx/dt)$^{2}$]x = 0 \end{center} has only periodic solutions. The application of standard perturbation methods, harmonic balance, and other approximation techniques all reach the conclusion that the angular frequency has a singularity for a finite value of the initial amplitude A, where the initial conditions are x(0) = A and dx(0)/dt = 0. Since a phase-space analysis demonstrates that such a singularity does not exist, we must seek other methods to give the required valid behavior for the angular frequency as a function or the initial amplitude. This presentation reports our work using a method based on a polar representation for the periodic solutions. We compare these results with a priori calculations and give an explanation as to why the earlier calculations were ``interpreted'' as being incorrect.