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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

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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

4. [4]	“Cornell Electron Storage Ring.” (2014), CLASSE: CESR. 2014 Cornell Laboratory for Accelerator-based Sciences and Education

5. [5]	Meller, R.E., personal communication to author RR on 12-27-13

6. [6]	Kevorkian, J. and Cole, J.D. (1981), Perturbation Methods in Applied Mathematics, Applied Mathematical Sciences, Vol. 34, Springer.

7. [7]	Rand, R.H. (2012), Lecture Notes in Nonlinear Vibrations, Published on-line by The Internet-First University Press.

8. [8]	Routh, E.J.(1877), A treatise on the stability of a given state of motion, particularly steady motion, Macmillan, London, U.K.

9. [9]	MATLAB’s reference on dde23. http://www.mathworks.com/help/matlab/ref/dde23.html