First-Order Averaging Principles for Maps with Applications to Accelerator Beam Dynamics

For slowly evolving, discrete-time–dependent systems of difference equations (iterated maps), we believe that the simplest means of demonstrating the validity of the averaging method at first order is by way of a lemma that we call the Besjes inequality. In this paper, we develop the Besjes inequality for identity maps with perturbations that are (i) at low-order resonance (periodic with short period) and (ii) far from low-order resonance in discrete time. We use these inequalities to prove corresponding first-order averaging principles, together with a principle of adiabatic invariance on extended timescales, and we generalize and apply these mathematical results to model problems in accelerator beam dynamics and to the Hénon map.