Simple Hamiltonian approach to describe large-amplitude and higher-order parametric resonances

Parametric resonance is a complex phenomenon that touches many aspects of scientific and technical society, but is still not well understood because of the intensive calculations required to describe the behavior. Thus the importance of developing simple mathematical approaches to describe parametric resonance cannot be overstated. Here a consistent theory of the parametric resonance of a harmonic oscillator under any periodic frequency modulation is constructed. Using a Hamiltonian approach and resonance approximation, simple equations were derived and critical amplitudes for all parametric resonance orders were obtained for any periodic modulation function. The theory agrees with the well-known result for the main resonance and gives correct power dependence on damping. In addition, the theory qualitatively predicts behavior at large modulations. This simplified approach revealed unique features-"safety windows" at large modulation amplitudes where parametric resonance does not occur which were then qualitatively confirmed with numerical simulations. The Hamiltonian approach should serve as a framework that a greater understanding of parametric resonance at large amplitudes and higher orders can be built upon.