Cubic velocity anti-damping
= Cubic velocity anti-damping
{title2=$R_T=3R^3/8$}
For $y''+y=\varepsilon(y')^3$ with positive $\varepsilon$, the oscillator <energy> increases at rate $\varepsilon(y')^4$. The <amplitude-phase equations> give $R(T)=R_0(1-3R_0^2T/4)^{-1/2}$. The <weakly nonlinear expansion> fails when the growing amplitude makes $\varepsilon R^2$ large, so its formal finite-time divergence does not itself control the exact late-time trajectory.