 Cubic velocity anti-damping

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