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Multi-stability and almost periodic solutions of a class of recurrent neural networks. (English) Zbl 1136.34311

The paper studies a class of reccurent neural networks described by the equations

${\stackrel{˙}{x}}_{i}\left(t\right)=-{a}_{i}{x}_{i}\left(t\right)+\sum _{j=1}^{n}{w}_{ij}f\left({x}_{j}\left(t\right)\right)+{c}_{i}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}f\left(x\right)\in \left(-1,\phantom{\rule{0.166667em}{0ex}}1\right)\phantom{\rule{1.em}{0ex}}i=1,\cdots ,n·$

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.

##### MSC:
 34C27 Almost and pseudo-almost periodic solutions of ODE 34D20 Stability of ODE 92B20 General theory of neural networks (mathematical biology)
##### Keywords:
almost periodic solutions; stability; neural networks.
Matlab