We study the Van der Pol oscillator \(\ddot{x} - \mu(1-x^2)\dot{x} + x = Fx\cos\omega t\), in which the external excitation is multiplied by the state, so that the forcing acts parametrically and vanishes at the origin. Written as a Van der Pol–Mathieu oscillator, the equation admits a complete analysis of its 2:1 resonance, and every result below is proved. First, the trivial solution is unstable for every \(F\) and \(\omega\) when \(\mu \gt 0\), with Floquet exponents \(\mu/2 \pm s\), where \(s\) is the exponent of an associated Mathieu equation; the sum of the exponents of any periodic orbit equals \(\mu(1-\langle x^2\rangle)\), and every periodic orbit satisfies an exact power balance. Second, the first-order averaged flow is triangular, and the substitution \(u=a^{-2}\) makes its amplitude equation linear; this yields global statements. Inside the tongue \(|\omega-2| \lt F/2\) all solutions except one are attracted to a subharmonic orbit of amplitude \(a_+ = 2\sqrt{1+\mathscr{D}/(2\mu)} \gt 2\), \(\mathscr{D} = \sqrt{F^2-4(\omega-2)^2}\), while a saddle orbit of amplitude \(a_- = 2\sqrt{1-\mathscr{D}/(2\mu)}\) exists if and only if \(\mathscr{D} \lt 2\mu\); both merge with the origin on the hyperbola \(F^2 = 4(\omega-2)^2+4\mu^2\). Outside the tongue a unique attracting orbit of the slow flow, with closed-form amplitude envelope, transverse rate \(\mu\) and phase-rotation frequency \(\kappa/2\), is obtained. Third, by averaging and normal hyperbolicity these objects persist in the full equation as a hyperbolic subharmonic periodic orbit, a saddle periodic orbit and an attracting invariant torus, respectively. Dimensional design rules follow. An appendix, clearly separated from the proofs, confirms the first-order predictions and, for strong forcing, indicates chaos; two precise open problems are formulated.