Analytical estimations of limit cycle amplitude for delay-differential equations

Tamás G. Molnár, T. Insperger, G. Stépán

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The amplitude of limit cycles arising from Hopf bifurcation is estimated for nonlinear delay-differential equations by means of analytical formulas. An improved analytical estimation is introduced, which allows more accurate quantitative prediction of periodic solutions than the standard approach that formulates the amplitude as a square-root function of the bifurcation parameter. The improved estimation is based on special global properties of the system: the method can be applied if the limit cycle blows up and disappears at a certain value of the bifurcation parameter. As an illustrative example, the improved analytical formula is applied to the problem of stick balancing.

Original languageEnglish
Article number77
JournalElectronic Journal of Qualitative Theory of Differential Equations
Publication statusPublished - 2016



  • Center manifold reduction
  • Delay-differential equation
  • Hopf bifurcation
  • Limit cycle
  • Normal form theory

ASJC Scopus subject areas

  • Applied Mathematics

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