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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
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