Analysis of Hopf bifurcations in differential equations with state-dependent delays via multiple scales method

Lijun Pei*, Shuo Wang, Marian Wiercigroch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


In this paper, a multiple scales method (MMS) is employed to analyze Hopf bifurcations in differential equations with two linearly state-dependent time delays. Firstly, the linear stability of the linearized equation near the only equilibrium (the trivial equilibrium) is performed analytically. Then, the case for which the coefficients of the delayed terms are small, the method of multiple scales (MMS) bypassing the need to use center manifold reduction allows the normal form to be easily obtained. Furthermore, the stability and bifurcation analysis are undertaken for the normal form to determine the types of the Hopf bifurcation. The proposed method can not only determine the direction of Hopf bifurcation but also its type. The numerical simulation results agree well with the analytical predictions. This suggests that the MMS employed in this paper provides a simple, accurate and effective means of analyzing Hopf bifurcations in the state-dependent delayed differential equations.

Original languageEnglish
Pages (from-to)789-801
Number of pages13
JournalJournal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM)
Issue number5
Early online date21 Dec 2017
Publication statusPublished - 31 May 2018


  • formal linearization
  • Hopf bifurcation
  • multiple scales method
  • stability analysis
  • state-dependent delays

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