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**Hopf bifurcation analysis in a tri-neuron network with time delay.**
*(English)*
Zbl 1149.34044

The paper studies the following three coupled delay differential equations (tri-neuron network)
\[
\begin{aligned} \dot{x}_{1}(t) & = \mu x_{1}(t) +f_{11}(x_{1}(t-\tau)) +f_{12}(x_{2}(t-\tau)) +f_{13}(x_{3}(t-\tau)),\\ \dot{x}_{2}(t) & = \mu x_{2}(t) +f_{21}(x_{1}(t-\tau)) +f_{22}(x_{2}(t-\tau)) +f_{23}(x_{3}(t-\tau)),\\ \dot{x}_{3}(t) & = \mu x_{3}(t) +f_{31}(x_{1}(t-\tau)) +f_{32}(x_{2}(t-\tau)) +f_{33}(x_{3}(t-\tau)),\end{aligned}
\]
where \(\tau>0\). Considering delay \(\tau\) as a parameter, the paper investigates existence and direction of Hopf bifurcations as well as stability of bifurcating periodic solutions. The group of conditions is given for the system to have multiple periodic solutions when the delay becomes sufficiently large.

Reviewer: Sergiy Yanchuk (Berlin)

### MSC:

34K18 | Bifurcation theory of functional-differential equations |

34K20 | Stability theory of functional-differential equations |

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

34K13 | Periodic solutions to functional-differential equations |

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\textit{D. Fan} and \textit{J. Wei}, Nonlinear Anal., Real World Appl. 9, No. 1, 9--25 (2008; Zbl 1149.34044)

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