Reading a Paper 2026, 10 - Non-Floquet Oscillations of a Parametrically Driven Rigid Planar Pendulum

Planar pendulums are classical systems with 1 d.o.f. usually considered to be driven externally to sustain the oscillations against disspative forces. Its angular frequency is modulated periodically and experiences parametric resonances. Given a planar pendulum in the xy-plane under influence of gravity, and additional sinusoidal vibrations of the pivot with amplitude a and angular frequency ω, the Lagrangian is

(Eq. 1). The dissipation force F_dis = -γv is velocity-dependent and in fact the negative derivative of the Rayleigh dissipation function.

Define the rest state at θ = 0. The change of perturbations under disturbance is noted as Θ and Ψ, which introduce a time-periodic 2 × 2 square matrix in the modified Mathieu equation

(Eqs. 3-5) where j = ±1. The solution can be expressed as a product of an exponential term e^μτ and a periodic function F(τ), with α = s + iαΩ. The angular displacement is real iff α is a multiple of 1/2. The remaining values are non-physical. Numerical computation of the Floquet multipliers uses through integrating the dynamical system for one period of driving with two different initial conditions and solving for τ = T. The eigenvalues of the monodromy matrix B(T) = [Φ₁(T), Φ₂]. The product of the Floquet multipliers needs to be equal to unity for β = 0 and e^-2β otherwise. Stability zones in the Ω-A plane is computed using a technique discussed for a parametrically driven double pendulum (3). For a damped planar pendulum, the damping term changes exponentially over time except at the boundaries of instability zones, where they remain static. It's written in autonomous form

(Eq. 6) as a dynamical system, integrated using standard 4th-order Runge-Kutta method with small initial values for θ and ψ around fixed points. Z₀ is assumed 0.

Chapter 4. is dedicated to the results of the simulations, associating the power-spectra with the phase diagrams. The power-spectrum has general shape, which gans more local peaks, the more heavily perturbed the oscillation is. In the cases of non-Floquet oscillations, the sum of the frequencies corresponds to the two largest peaks adding to the driving frequency. This is analogous to the energy conservation condition of SPDC in quantum optics (5.)