Parametric Entrainment and Amplitude Enhancement in a Van der Pol Oscillator with State-Multiplicative Periodic Forcing: A Rigorous Averaging Analysis

We study the Van der Pol oscillator ¨ 𝑥 −𝜇⁢(1 −𝑥2)⁢ ˙ 𝑥 +𝑥 =𝐹⁡𝑥⁢cos⁡𝜔⁢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 𝜔 when 𝜇 >0, with Floquet exponents 𝜇/2 ±𝑠, where 𝑠 is the exponent of an associated Mathieu equation; the sum of the exponents of any periodic orbit equals 𝜇⁢(1 −⟨𝑥2⟩), 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 |𝜔 −2| <𝐹⁡/2 all solutions except one are attracted to a subharmonic orbit of amplitude 𝑎+ =2⁢√1+𝒟/(2⁢𝜇) >2, 𝒟 =√𝐹2−4⁢(𝜔−2)2, while a saddle orbit of amplitude 𝑎− =2⁢√1−𝒟/(2⁢𝜇) exists if and only if 𝒟 <2⁢𝜇; both merge with the origin on the hyperbola 𝐹2 =4⁢(𝜔−2)2 +4⁢𝜇2. Outside the tongue a unique attracting orbit of the slow flow, with closed-form amplitude envelope, transverse rate 𝜇 and phase-rotation frequency 𝜅/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.