The Dynamics of One Way Coupling in a System of Nonlinear Mathieu Equations

Abstract

Background: This paper extends earlier research on the dynamics of two coupled Mathieu equations by introducing nonlinear terms and focusing on the effect of one-way coupling. The studied system of n equations models the motion of a train of n particle bunches in a circular particle accelerator. Objective: The goal is to determine (a) the system parameters which correspond to bounded motion, and (b) the resulting amplitudes of vibration for parameters in (a). Method: We start the investigation by examining two coupled equations and then generalize the results to any number of coupled equations. We use a perturbation method to obtain a slow flow and calculate its nontrivial fixed points to determine steady state oscillations. Results: The perturbation method reveals the existence of an upper bound on the amplitude of steady state oscillations. Conclusion: The model predicts how many bunches may be included in a train before instability occurs.