A strict amplitude lower bound for the van der Pol equation and extensions to symmetric Liénard systems

We prove that the amplitude of the unique limit cycle of the van der Pol equation $\ddot{x}+μ(x^2-1)\dot{x}+x=0$ is strictly greater than $2$ for every $μ>0$. The proof combines orbit comparison in the relaxation oscillation regime, trigonometric comparison curves in the weakly nonlinear regime, and