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Some basic remarks on eigenmode expansions of time-delay dynamics

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  • Amann, Andreas
  • Schöll, Eckehard
  • Just, Wolfram

Abstract

We describe in elementary terms how eigenmode expansions can be used to deal with differential-difference equations. As particular applications we present the full analytical solution of linear stochastic time-delay systems and the weakly nonlinear analysis of nonlinear differential-difference equations in the limit of large time delay. Our exposition is essentially based on an explicit analytical expression for the linear spectrum in terms of the Lambert W-function and on the explicit formula for the eigenfunctions of the adjoint equation.

Suggested Citation

  • Amann, Andreas & Schöll, Eckehard & Just, Wolfram, 2007. "Some basic remarks on eigenmode expansions of time-delay dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 373(C), pages 191-202.
    • Handle: RePEc:eee:phsmap:v:373:y:2007:i:c:p:191-202
    DOI: 10.1016/j.physa.2005.12.073
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    References listed on IDEAS

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    1. Kawasaki, Kyozi & Ohta, Takao, 1982. "Kink dynamics in one-dimensional nonlinear systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 116(3), pages 573-593.
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    Cited by:

    1. Lucas Wetzel & David J Jörg & Alexandros Pollakis & Wolfgang Rave & Gerhard Fettweis & Frank Jülicher, 2017. "Self-organized synchronization of digital phase-locked loops with delayed coupling in theory and experiment," PLOS ONE, Public Library of Science, vol. 12(2), pages 1-21, February.
    2. Warburton, Roger D.H. & Hodgson, J.P.E. & Nielsen, E.H., 2014. "Exact solutions to the supply chain equations for arbitrary, time-dependent demands," International Journal of Production Economics, Elsevier, vol. 151(C), pages 195-205.

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