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**Multistability in networks of weakly coupled bistable units.**
*(English)*
Zbl 0891.34060

Summary: We study the stationary states of networks consisting of weakly coupled bistable units. We prove the existence of a high multiplicity of stable steady states in networks with very general inter-unit dynamics. We present a method for estimating the critical coupling strength below which these stationary states persist in the network. In some cases, the presence of time-independent localized states in the system can be regarded as a ‘propagation failure’ phenomenon. We analyse this type of behaviour in the case of diffusive networks whose elements are described by one or two variables and give concrete examples.

### MSC:

34D30 | Structural stability and analogous concepts of solutions to ordinary differential equations |

94C05 | Analytic circuit theory |

34D20 | Stability of solutions to ordinary differential equations |

34C99 | Qualitative theory for ordinary differential equations |

93D99 | Stability of control systems |

34D10 | Perturbations of ordinary differential equations |

70K20 | Stability for nonlinear problems in mechanics |

### Keywords:

weakly coupled bistable units; high multiplicity of stable steady states; diffusive networks
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\textit{R. S. MacKay} and \textit{J. A. Sepulchre}, Physica D 82, No. 3, 243--254 (1995; Zbl 0891.34060)

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