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

Alexander Bernstein1, Richard Rand2, *, Robert Meller3
1 Center for Applied Mathematics, Cornell University, Ithaca, NY 14850, USA
2 Dept. of Mathematics and Dept. of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14850, USA
3 Cornell Lab for Accelerator-based Sciences and Education, Cornell University, Ithaca, NY 14850, USA

© 2018 Bernstein et al.

open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: (https://creativecommons.org/licenses/by/4.0/legalcode). This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

* Address correspondence to this authors at the Dept. of Mathematics, Cornell University, Ithaca, NY 14850, USA; Tel: 607 255 8198; E-mails: rhr2@cornell.edu, rrand1@twcny.rr.com

## 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.

Keywords: Parametric vibrations, Coupled oscillators, Mathieu’s equation, Synchrotron, Bifurcation theory, Perturbation methods.