Multi-stability and almost periodic solutions of a class of recurrent neural networks.

*(English)*Zbl 1136.34311The paper studies a class of reccurent neural networks described by the equations

\[ \dot x_i(t)=-a_i x_i(t)+\sum_{j=1}^n w_{ij} f(x_j(t))+c_i\,,\quad f(x)\in (-1,\,1)\quad i=1,\dots,n. \]

Using Lyapunov functions, a sufficient condition for the complete stability is obtained. On this base applying the Mawhin coincidence degree theory, many sufficient conditions guaranteeing the existence of at least one almost periodic solution are obtained. These conditions are derived for an arbitrary activation function \(f\). Few simulations done by Matlab illustrate that the simulation results fit well the theoretic analysis.

\[ \dot x_i(t)=-a_i x_i(t)+\sum_{j=1}^n w_{ij} f(x_j(t))+c_i\,,\quad f(x)\in (-1,\,1)\quad i=1,\dots,n. \]

Using Lyapunov functions, a sufficient condition for the complete stability is obtained. On this base applying the Mawhin coincidence degree theory, many sufficient conditions guaranteeing the existence of at least one almost periodic solution are obtained. These conditions are derived for an arbitrary activation function \(f\). Few simulations done by Matlab illustrate that the simulation results fit well the theoretic analysis.

Reviewer: Ivan Ginchev (Varese)

##### MSC:

34C27 | Almost and pseudo-almost periodic solutions to ordinary differential equations |

34D20 | Stability of solutions to ordinary differential equations |

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

##### Software:

Matlab
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\textit{Y. Liu} and \textit{Z. You}, Chaos Solitons Fractals 33, No. 2, 554--563 (2007; Zbl 1136.34311)

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