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Dynamics of an inertial two-neuron system with time delay. (English) Zbl 1183.68485

Summary: We considered a delayed differential equation modeling two-neuron system with both inertial terms and time delay. By analyzing the distribution of the eigenvalues of the corresponding transcendental characteristic equation of its linearized equation, local stability criteria are derived for various model parameters and time delays. By choosing the time delay as a bifurcation parameter, the model is found to undergo a sequence of Hopf bifurcations. Furthermore, the direction and the stability of the bifurcating periodic solutions are determined by using normal form theory and the center manifold theorem. Also, a resonant codimension-two bifurcation is found to occur in this model. Some numerical examples are finally given for justifying the theoretical results. Chaotic behavior of this inertial two-neuron system with time delay is found also through numerical simulations, in which some phase plots, waveform plots, power spectra and Lyapunov exponents are computed and presented.

MSC:

68T05 Learning and adaptive systems in artificial intelligence
34K60 Qualitative investigation and simulation of models involving functional-differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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