Coupled Parametrically Driven Modes in Synchrotron Dynamics

  • Alexander BernsteinEmail author
  • Richard Rand
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)


This work concerns the dynamics of a type of particle accelerator called a synchrotron, in which particles are made to move in nearly circular orbits of large radius. The stability of the transverse motion of such a rotating particle may be modeled as being governed by Mathieu’s equation. For a train of two such particles the equations of motion are coupled due to plasma interactions and resistive wall coupling effects.In this paper we study a system consisting of a train of two such particles which is modeled as two coupled nonlinear Mathieu equations with delay coupling. In particular we investigate the stability of two coupled parametrically forced linear normal modes.


Parametric excitation Coupled oscillators Mathieu equation Stability analysis Synchrotron 



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.


  1. 1.
    Morrison TM, Rand RH (2007) Resonance in the delayed nonlinear Mathieu equation. Nonlinear Dyn 341–352. doi: 10.1007/s11071-006-9162-5
  2. 2.
    Rand RH (2012) Lecture notes in nonlinear vibrations. The Internet-First University Press, New York.

Copyright information

© The Society for Experimental Mechanics, Inc. 2016

Authors and Affiliations

  1. 1.Center for Applied MathematicsCornell UniversityIthacaUSA
  2. 2.Department of MathematicsCornell UniversityIthacaUSA
  3. 3.Department of Mechanical and Aerospace EngineeringCornell UniversityIthacaUSA

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