Collective dynamics of duffing oscillators: Model for dipole ripple

The emittance growth due to power supply ripple in dipoles is modeled by the collective behavior of forced Duffing oscillators. The method of averaging reduces the problem to an autonomous system. A coarse grained long time limit of the phase space density and the rate of approach to this limit are discussed in terms of the autonomous system. The equilibrium density, the rate of approach to equilibrium and the equilibrium values depend crucially on the detuning parameter. We find the frequencies which lead to the largest emittance growth in three different forcing regimes and also characterize the dependence of emittance growth on forcing amplitude in these regimes