Stability and Neimark-Sacker bifurcation of numerical discretization of delay differential equations.

*(English)*Zbl 1198.65107Summary: A kind of a discrete delay model obtained by Euler method is investigated. Firstly, the linear stability of the model is studied. It is found that there exist Neimark-Sacker bifurcations when the delay passes a sequence of critical values. Then the explicit algorithm for determining the direction and stability of the Neimark-Sacker bifurcations are derived by using the normal form theory and center manifold theorem. Finally, computer simulations are provided to illustrate the analytical results found.

Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

##### MSC:

65L03 | Numerical methods for functional-differential equations |

65P30 | Numerical bifurcation problems |

37G99 | Local and nonlocal bifurcation theory for dynamical systems |

34K18 | Bifurcation theory of functional-differential equations |

34K20 | Stability theory of functional-differential equations |

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\textit{Z. He} et al., Chaos Solitons Fractals 41, No. 4, 2010--2017 (2009; Zbl 1198.65107)

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