##
**Multistability of competitive neural networks with time-varying and distributed delays.**
*(English)*
Zbl 1167.34383

Summary: For two classes of general activation functions, we investigate the multistability of competitive neural networks with time-varying and distributed delays. By formulating parameter conditions and using inequality technique, several novel delay-independent sufficient conditions ensuring the existence of \(3^N\) equilibria and exponential stability of \(2^N\) equilibria are derived. In addition, estimations of positively invariant sets and basins of attraction for these stable equilibria are obtained. Two examples are given to show the effectiveness of our theory.

### MSC:

34K20 | Stability theory of functional-differential equations |

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

37N25 | Dynamical systems in biology |

PDFBibTeX
XMLCite

\textit{X. Nie} and \textit{J. Cao}, Nonlinear Anal., Real World Appl. 10, No. 2, 928--942 (2009; Zbl 1167.34383)

Full Text:
DOI

### References:

[1] | Chua, L. O.; Yang, L., Cellular neural networks: Theory, IEEE Trans. Circuits Syst., 35, 1257-1272 (1988) · Zbl 0663.94022 |

[2] | Hopfield, J., Neurons with graded response have collective computational properties like those of two state neurons, Proc. Natl. Acad. Sci. USA, 81, 3088-3092 (1984) · Zbl 1371.92015 |

[3] | Morita, M., Associative memory with non-monotone dynamics, Neural Netw., 6, 115-126 (1993) |

[4] | Foss, J.; Longtin, A.; Mensour, B.; Milton, J., Multistability and delayed recurrent loops, Phys. Rev. Lett., 76, 708-711 (1996) |

[5] | Meyer-Baese, A.; Ohl, F.; Scheich, H., Singular perturbation analysis of competitive neural networks with different time-scales, Neural Comput., 8, 1731-1742 (1996) |

[6] | Cao, J.; Wang, J., Asymptotic stability of a general class of recurrent neural networks with time-varying delays, IEEE Trans. Circuits Syst. I, 50, 1, 34-44 (2003) · Zbl 1368.34084 |

[7] | Cao, J.; Song, Q., Stability in Cohen-Grossberg-type bidirectional associative memory neural networks with time-varying delays, Nonlinearity, 19, 1601-1607 (2006) · Zbl 1118.37038 |

[8] | Yuan, K.; Cao, J., Exponential stability and periodic solutions of fuzzy cellular neural networks with time-varying delays, Neurocomputing, 69, 1619-1627 (2006) |

[9] | Zhang, Q.; Wei, X.; Xu, J., Stability analysis for cellular neural networks with variable delays, Chaos Solitons Fractals, 28, 331-336 (2006) · Zbl 1084.34068 |

[10] | Liu, B.; Huang, L., Existence and stability of almost periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays, Phys. Lett. A, 349, 177-186 (2006) |

[11] | Song, Q.; Cao, J.; Zhao, Z., Periodic solutions and its exponential stability of reaction-diffusion recurrent neural networks with continuously distributed delays, Nonlinear Anal. Real World Appl., 7, 65-80 (2006) · Zbl 1094.35128 |

[12] | Jiang, H.; Teng, Z., Global exponential stability of cellular neural networks with time-varying coefficients and delays, Neural Netw., 17, 10, 1415-1425 (2004) · Zbl 1068.68121 |

[13] | Jiang, H.; Teng, Z., A new criterion on the global exponential stability for cellular neural networks with multiple time-varying delays, Phys. Lett. A, 338, 461-471 (2005) · Zbl 1136.34338 |

[14] | Meyer-Baese, A.; Pilyugin, S.; Chen, Y., Global exponential stability of competitive neural networks with different time scales, IEEE Trans. Neural Netw., 14, 3, 716-719 (2003) |

[15] | Meyer-Baese, A.; Pilyugin, S.; Wismuller, A.; Foo, S., Local exponential stability of competitive neural networks with different time scales, Eng. Appl. Artifical Intell., 17, 227-232 (2004) |

[16] | Lu, H.; Shun-ichi, A., Global exponential stability of multitime scale competitive neural networks with nonsmooth functions, IEEE Trans. Neural Netw., 17, 5, 1152-1164 (2006) |

[17] | Lu, H.; He, Z., Global exponential stability of delayed competitive neural networks with different time scales, Neural Netw., 18, 243-250 (2005) · Zbl 1078.68126 |

[18] | Mao, Y., Multistability of competitive neural networks with different time scales, (Communications, Circuits and Systems, 2005. Proceedings. 2005 International Conference on, vol. 2 (2005)), 939-943 |

[19] | Cheng, C. Y.; Lin, K. H.; Shih, C. W., Multistability in recurrent neural networks, SIAM J. Appl. Math., 66, 4, 1301-1320 (2006) · Zbl 1106.34048 |

[20] | Cheng, C. Y.; Lin, K. H.; Shih, C. W., Multistability and convergence in delayed neural networks, Physica D, 225, 61-74 (2007) · Zbl 1132.34058 |

[21] | Zeng, Z. G.; Wang, J., Multiperiodicity and exponential attractivity evoked by periodic external inputs in delayed cellular neural networks, Neural Comput., 18, 848-870 (2006) · Zbl 1107.68086 |

[22] | Kuang, J., Applied Inqualities (2004), Shandong Science and Technology Press, (in Chinese) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.