Two models for the parametric forcing of a nonlinear oscillator

Instabilities associated with 2:1 and 4:1 resonances of two models for the parametric forcing of a strictly nonlinear oscillator are analyzed. The first model involves a nonlinear Mathieu equation and the second one is described by a 2 degree of freedom Hamiltonian system in which the forcing is introduced by the coupling. Using averaging with elliptic functions, the threshold of the overlapping phenomenon between the resonance bands 2:1 and 4:1 (Chirikov’s overlap criterion) is determined for both models, offering an approximation for the transition from local to global chaos. The analytical results are compared to numerical simulations obtained by examining the Poincaré section of the two systems.