Set stability of Boolean networks via quotient mappings

The Boolean network is a powerful tool in describing, analyzing, and simulating the cellular networks. In this paper, we investigate stability of given subsets (or set stability) for Boolean networks which includes synchronization problem as a special case. First of all, based on the semi-tensor product of matrices and the algebraic expression of Boolean networks, we derive some necessary and sufficient conditions for set stability. Then we propose a quotient mapping method the main idea of which is to regard the largest invariant set contained in a pre-specified subset as a single point and then transfer the set stability problem of the original Boolean network to the stability of the multi-logical systems in the quotient space. Examples are also provided to explain the results obtained.