A Theoretical Estimate for the Frequency of the TNL Oscillator $\ddot x + x + x^{1/3}=0$

Truly nonlinear (TNL) oscillators have the property of having no linear approximation at the fixed-point of the modeling differential equation [1]. For a conservative oscillator this means that the fixed-point is a nonlinear center. Another feature of TNL oscillators is that none of the standard perturbation expansion procedures can be applied to calculate analytical approximations to the periodic or oscillator solutions [2]. Using the initial conditions $x(0)=A$ and $\dot x(0)=0$, we calculate the frequency $\omega (A)$ of the equation given in the title for small, $0