Nonlinear dynamics and chaos in fractional-order neural networks.

*(English)*Zbl 1254.34103Summary: Several topics related to the dynamics of fractional-order neural networks of Hopfield type are investigated, such as stability and multi-stability (coexistence of several different stable states), bifurcations and chaos. The stability domain of a steady state is completely characterized with respect to some characteristic parameters of the system, in the case of a neural network with ring or hub structure. These simplified connectivity structures play an important role in characterizing the network’s dynamical behavior, allowing us to gain insight into the mechanisms underlying the behavior of recurrent networks. Based on the stability analysis, we are able to identify the critical values of the fractional order for which Hopf bifurcations may occur. Simulation results are presented to illustrate the theoretical findings and to show potential routes towards the onset of chaotic behavior when the fractional order of the system increases.

##### MSC:

34K20 | Stability theory of functional-differential equations |

34K18 | Bifurcation theory of functional-differential equations |

34A08 | Fractional ordinary differential equations |

##### Keywords:

neural networks; fractional order; fractance; stability; multistability; Hopf bifurcation; chaos; strange attractor; ring; hub
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\textit{E. Kaslik} and \textit{S. Sivasundaram}, Neural Netw. 32, 245--256 (2012; Zbl 1254.34103)

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