Computation of double Hopf points for delay differential equations

Yingxiang Xu; Tingting Shi

Open Mathematics (2015)

  • Volume: 13, Issue: 1
  • ISSN: 2391-5455

Abstract

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Relating to the crucial problem of branch switching, the calculation of codimension 2 bifurcation points is one of the major issues in numerical bifurcation analysis. In this paper, we focus on the double Hopf points for delay differential equations and analyze in detail the corresponding eigenspace, which enable us to obtain the finite dimensional defining system of equations of such points, instead of an infinite dimensional one that happens naturally for delay systems. We show that the double Hopf point, together with the corresponding eigenvalues, eigenvectors and the critical values of the bifurcation parameters, is a regular solution of the finite dimensional defining system of equations, and thus can be obtained numerically through applying the classical iterative methods. We show our theoretical findings by a numerical example.

How to cite

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Yingxiang Xu, and Tingting Shi. "Computation of double Hopf points for delay differential equations." Open Mathematics 13.1 (2015): null. <http://eudml.org/doc/275838>.

@article{YingxiangXu2015,
abstract = {Relating to the crucial problem of branch switching, the calculation of codimension 2 bifurcation points is one of the major issues in numerical bifurcation analysis. In this paper, we focus on the double Hopf points for delay differential equations and analyze in detail the corresponding eigenspace, which enable us to obtain the finite dimensional defining system of equations of such points, instead of an infinite dimensional one that happens naturally for delay systems. We show that the double Hopf point, together with the corresponding eigenvalues, eigenvectors and the critical values of the bifurcation parameters, is a regular solution of the finite dimensional defining system of equations, and thus can be obtained numerically through applying the classical iterative methods. We show our theoretical findings by a numerical example.},
author = {Yingxiang Xu, Tingting Shi},
journal = {Open Mathematics},
keywords = {double Hopf points; delay differential equations; regular defining system of equations; iterative methods},
language = {eng},
number = {1},
pages = {null},
title = {Computation of double Hopf points for delay differential equations},
url = {http://eudml.org/doc/275838},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Yingxiang Xu
AU - Tingting Shi
TI - Computation of double Hopf points for delay differential equations
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - null
AB - Relating to the crucial problem of branch switching, the calculation of codimension 2 bifurcation points is one of the major issues in numerical bifurcation analysis. In this paper, we focus on the double Hopf points for delay differential equations and analyze in detail the corresponding eigenspace, which enable us to obtain the finite dimensional defining system of equations of such points, instead of an infinite dimensional one that happens naturally for delay systems. We show that the double Hopf point, together with the corresponding eigenvalues, eigenvectors and the critical values of the bifurcation parameters, is a regular solution of the finite dimensional defining system of equations, and thus can be obtained numerically through applying the classical iterative methods. We show our theoretical findings by a numerical example.
LA - eng
KW - double Hopf points; delay differential equations; regular defining system of equations; iterative methods
UR - http://eudml.org/doc/275838
ER -

References

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