Fermi resonance and its effect on the mean transition time and rate

Fermi resonance and its effect on the mean transition time and rate are studied. The necessary frequency ratio 1:2 for Fermi resonance to occur is explained by applying the deterministic averaging method to the two-dimensional conservative Pippard system, and a more frequent fluctuation of energy process due to Fermi resonance is shown by using the samples obtained from digital simulation of the stochastic Pippard system. In the case of weak coupling, the mean transition time of the reacting oscillator energy is evaluated for both nonresonance and Fermi resonance by using the standard (Stratonovich) stochastic averaging method. The theoretical results for the mean transition time in the case of Fermi resonance and nonresonance is then extended to the stochastic system with bistable potential, and the effects of frequency ratio and coupling coefficient on the mean reaction rate are analyzed. In the case of strong coupling, it is pointed out that the exciting oscillator and reacting oscillator move together like one oscillator and no Fermi resonance can occur. In this case, the mean transition rate of the system total energy is studied by using the stochastic averaging method for quasi-non-integrable Hamiltonian systems. All the theoretical results are confirmed through comparison with those from digital simulation, and the effect of Fermi resonance on the transition time and rate is discussed.