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.


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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