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**Hopf bifurcation and chaos in a single delayed neuron equation with non-monotonic activation function.**
*(English)*
Zbl 1012.92005

Summary: A simple neural network model with discrete time delays is investigated. The linear stability of this model is discussed by analyzing the associated characteristic transcendental equation. For the case with inhibitory influence from the past state, it is found that Hopf bifurcation occurs when this influence varies and passes through a sequence of critical values. The stability of bifurcating periodic solutions and the direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Chaotic behavior of a single delayed neuron equation with non-monotonously increasing transfer function has been observed in computer simulations. Some waveform diagrams, phase portraits, power spectra and plots of the largest Lyapunov exponent will also be given.

### MSC:

92B20 | Neural networks for/in biological studies, artificial life and related topics |

34K23 | Complex (chaotic) behavior of solutions to functional-differential equations |

34K18 | Bifurcation theory of functional-differential equations |

34K13 | Periodic solutions to functional-differential equations |

37N25 | Dynamical systems in biology |

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

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\textit{X. Liao} et al., Chaos Solitons Fractals 12, No. 8, 1535--1547 (2001; Zbl 1012.92005)

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### References:

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