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

Lijun Pei, Shuo Wang, Marian Wiercigroch

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

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)
Volume98
Issue number5
Early online date21 Dec 2017
DOIs
Publication statusPublished - 31 May 2018

Keywords

  • formal linearization
  • Hopf bifurcation
  • multiple scales method
  • stability analysis
  • state-dependent delays
  • TIME-DELAY
  • DYNAMICS
  • OSCILLATOR
  • STABILITY
  • SYSTEM
  • MODEL

Cite this

@article{488489a3e5304d32ad896a0e25da8d0e,
title = "Analysis of Hopf bifurcations in differential equations with state-dependent delays via multiple scales method",
abstract = "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.",
keywords = "formal linearization, Hopf bifurcation, multiple scales method, stability analysis, state-dependent delays, TIME-DELAY, DYNAMICS, OSCILLATOR, STABILITY, SYSTEM, MODEL",
author = "Lijun Pei and Shuo Wang and Marian Wiercigroch",
note = "The authors would like to acknowledge the financial support for this research via the National Natural Science Foundation of China (No. 11372282) and China Scholarship Council (CSC). They also thank the reviewers for their valuable reviews and kind suggestions.",
year = "2018",
month = "5",
day = "31",
doi = "10.1002/zamm.201700172",
language = "English",
volume = "98",
pages = "789--801",
journal = "Journal of Applied Mathematics and Mechanics / Zeitschrift f{\"u}r Angewandte Mathematik und Mechanik (ZAMM)",
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publisher = "WILEY-V C H VERLAG GMBH",
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TY - JOUR

T1 - Analysis of Hopf bifurcations in differential equations with state-dependent delays via multiple scales method

AU - Pei, Lijun

AU - Wang, Shuo

AU - Wiercigroch, Marian

N1 - The authors would like to acknowledge the financial support for this research via the National Natural Science Foundation of China (No. 11372282) and China Scholarship Council (CSC). They also thank the reviewers for their valuable reviews and kind suggestions.

PY - 2018/5/31

Y1 - 2018/5/31

N2 - 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.

AB - 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.

KW - formal linearization

KW - Hopf bifurcation

KW - multiple scales method

KW - stability analysis

KW - state-dependent delays

KW - TIME-DELAY

KW - DYNAMICS

KW - OSCILLATOR

KW - STABILITY

KW - SYSTEM

KW - MODEL

U2 - 10.1002/zamm.201700172

DO - 10.1002/zamm.201700172

M3 - Article

VL - 98

SP - 789

EP - 801

JO - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM)

JF - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM)

SN - 0044-2267

IS - 5

ER -