Semi-Resonant Interactions and Frequency Dividers

A theory describing the behavior of a system as it evolves slowly through internal nonlinear resonances is presented. The energy sharing process is seen to be quite complex as it depends crucially on both nonlinear and frequency detuning effects. Two phenomena are discussed in detail although the general ideas are applicable to many situations. Firstly we examine the interaction between the quadratically coupled oscillators whose natural frequencies are in the ratio 2:1 for a limited period of time. Such a system is shown to be an extremely useful switching device. Secondly we examine the time dependent Duffing equation and find that smooth forward and reverse transitions occur without the presence of dissipation.