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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) $$\align \dot{x}_{1}(t) & = \mu x_{1}(t) +f_{11}(x_{1}(t-\tau)) +f_{12}(x_{2}(t-\tau)) +f_{13}(x_{3}(t-\tau)),\\ \dot{x}_{2}(t) & = \mu x_{2}(t) +f_{21}(x_{1}(t-\tau)) +f_{22}(x_{2}(t-\tau)) +f_{23}(x_{3}(t-\tau)),\\ \dot{x}_{3}(t) & = \mu x_{3}(t) +f_{31}(x_{1}(t-\tau)) +f_{32}(x_{2}(t-\tau)) +f_{33}(x_{3}(t-\tau)),\endalign$$ 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.

MSC:
34K18Bifurcation theory of functional differential equations
34K20Stability theory of functional-differential equations
34K60Qualitative investigation and simulation of models
34K13Periodic solutions of functional differential equations
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References:
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