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 $\ddot{x}+\mu (x^2-1)\dot{x}+x=0$ is strictly greater than $2$ for every $\mu >0$. The proof combines orbit comparison in the relaxation oscillation regime, trigonometric comparison curves in the weakly nonlinear regime, and polynomial energy estimates in the intermediate regime. The polynomial sign conditions arising in the latter two regimes are certified using exact rational arithmetic and Bernstein coefficients. The orbit comparison method also applies to a class of symmetric Liénard systems. Under suitable monotonicity assumptions, we establish a sufficient criterion for the limit cycle amplitude to exceed a critical value determined by the damping function.