Explore parametric resonance via the Mathieu equation: adjust modulation frequency and depth, observe the Ince-Strutt stability diagram
Parametric resonance occurs when a system parameter (rather than an external force) is modulated periodically. For an oscillator with natural frequency ω₀, modulating the spring constant at frequency Ω produces the Mathieu equation: d²x/dt² + 2γ(dx/dt) + ω₀²[1 + ε·cos(Ωt)]·x = 0. Unlike forced resonance where the driving frequency matches ω₀, parametric resonance is strongest when Ω ≈ 2ω₀ — twice the natural frequency. Small perturbations grow exponentially as x(t) ∝ exp(σt), where σ depends on ε and γ. The instability threshold is ε > 4γ/ω₀ for the principal zone. The Ince-Strutt diagram maps stable and unstable regions in the (ε, Ω/2ω₀) plane.
The Mathieu equation is a linear ODE with periodic coefficients. Floquet theory states solutions have the form x(t) = e^(σt)·p(t), where p(t) is T-periodic and σ is the Floquet exponent. Instability occurs when Re(σ) > 0. Numerically, we integrate the monodromy matrix M over one period with ICs (1,0) and (0,1), then compute the Floquet multipliers. The condition ρ(M) > 1, where ρ is the spectral radius, indicates instability and also applies with damping. The Ince-Strutt diagram is constructed by evaluating this condition over a grid of (ε, Ω/2ω₀). Multiple instability tongues appear at Ω/2ω₀ = 1/n for n = 1, 2, 3…, with n=1 being the widest.
Child on a swing: pumping legs at twice the swing frequency changes the effective pendulum length. Paul ion trap: RF electric fields create parametric confinement. Optical parametric oscillator (OPO): nonlinear crystal converts pump photons at Ω into signal/idler near ω₀ when Ω ≈ 2ω₀. MEMS mirrors: parametric excitation amplifies torsional oscillations. Faraday waves: vertically vibrated fluid forms standing waves at 2× the surface wave frequency. Parametric amplifiers: low-noise amplification in superconducting circuits and optics.
Start with Resonance: Ω/2ω₀ = 1.0 places you at the center of the principal instability tongue. Press Animate to see exponential amplitude growth. The stability diagram marks your position inside the red unstable zone. Switch to Off-Resonance: the crosshair moves to a stable (blue) region and amplitude decays. Try Threshold: just at the instability edge. Increase ε slightly to push into the unstable zone. Deep Modulation shows strong exponential growth. Subharmonic shows the second instability zone at Ω/2ω₀ = 0.5. Adjust damping γ to see instability regions shrink — the diagram updates automatically.