ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Delay-Coupled Mathieu Equations in Synchrotron Dynamics

Journal of Applied Nonlinear Dynamics 5(3) (2016) 337--348 | DOI:10.5890/JAND.2016.09.006

Alexander Bernstein; Richard Rand

$^{1}$ Center for Applied Mathematics, Cornell University

$^{2}$ Dept. of Mathematics and Dept. of Mechanical and Aerospace Engineering, Cornell University

Abstract

This paper investigates the dynamics of two coupled Mathieu equations. The coupling functions involve both delayed and nondelayed terms. We use a perturbation method to obtain a slow flow which is then studied using Routh-Hurwitz stability criterion. Analytic results are shown to compare favorably with numerical integration. The numerical integrator, DDE23, is shown to inadvertently add damping. It is found that the nondelayed coupling parameter plays a significant role in the system dynamics. We note that our interest in this problem comes from an application to the design of nuclear accelerators.

Acknowledgments

The authors wish to thank their colleagues J. Sethna, D. Rubin, D. Sagan and R. Meller for introducing us to the dynamics of the Synchrotron.

References

1.  [1] Hsu, C.S. (1961), On a restricted class of coupled Hill’s equations and some applications, Journal of Applied Mechanics, 28 (4), Series E, 551.
2.  [2] Bernstein, A. and Rand, R.H. (2016), Coupled Parametrically Driven Modes in Synchrotron Dynamics. Chapter 8, pp.107–112 in Nonlinear Dynamics, Volume 1: Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, G. Kerschen, editor, Springer.
3.  [3] Morrison, T.M. and Rand, R.H. (2007), 2:1 Resonance in the delayed nonlinear Mathieu equation, Nonlinear Dynamics, 341–352. doi:10.1007/s11071-006-9162-5
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9.  [9] MATLAB’s reference on dde23. http://www.mathworks.com/help/matlab/ref/dde23.html