Parametric excitation of a detuned spherical pendulum

The weakly non-linear response of a slightly detuned spherical pendulum with natural frequencies ω₁ and ω₂, ω₁ ≈ ω₂, subject to parametric excitation at frequency 2ω, is analyzed in the parametric domain ω² - 1/2; (ω² ₁ + ω² ₂) = O (εω²), where ε is a measure of the excitation. There is a finite neighbourhood of ω² = ω² ₂, bounded by Hopf-bifurcation points, in which no stable harmonic motion is possible and the motion is either a periodically modulated sinusoid (limit cycle) or chaotic, depending on the frequency offset and dissipation. The analytical predictions are supported by numerical integration, which also reveals that the parametric domains of limit cycles and chaotic motions overlap those of stable harmonic motion. The system provides a model of mode competition in a system with a discrete, infinite spectrum, e.g., Faraday waves.