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ALGEBRAIC APPROACH TO DYNAMICS OF MULTIVALUED NETWORKS

ZHIQIANG LI

Key Laboratory of Systems and Control, AMSS, Chinese Academy of Sciences, Beijing 100190, P. R. China

Address for Correspondence: No. 55, Zhongguancun East Road, Haidian District, Beijing 100190, P. R. China.

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DAIZHAN CHENG

Key Laboratory of Systems and Control, AMSS, Chinese Academy of Sciences, Beijing 100190, P. R. China

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https://doi.org/10.1142/S0218127410025892Cited by:96 (Source: Crossref)

Abstract

Using semi-tensor product of matrices, a matrix expression for multivalued logic is proposed, where a logical variable is expressed as a vector, and a logical function is expressed as a multilinear mapping. Under this framework, the dynamics of a multivalued logical network is converted into a standard discrete-time linear system. Analyzing the network transition matrix, easily computable formulas are obtained to show (a) the number of equilibriums; (b) the numbers of cycles of different lengths; (c) transient period, the minimum time for all points to enter the set of attractors, respectively. A method to reconstruct the logical network from its network transition matrix is also presented. This approach can also be used to convert the dynamics of a multivalued control network into a discrete-time bilinear system. Then, the structure and the controllability of multivalued logical control networks are revealed.

Keywords:

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