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: ( 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:,



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.


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).


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.


The perturbation method reveals the existence of an upper bound on the amplitude of steady state oscillations.


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.