Graphic Requirements for Multiple Attractive Cycles in Boolean Dynamical Systems

Jian-Lang Dong

Abstract


E. Remy, P. Ruet and D. Thieffry have proved a Boolean version of Thomas' conjecture: if a map $F$ from $\{0,1\}^{n}$ to itself has several fixed points, then there exists a positive circuit in the corresponding interaction graph. In this paper, we prove that the presence of a positive circuit in a local interaction graph is also a necessary condition for the presence of several attractive cycles in the Boolean synchronous dynamics.

Keywords


Gene network; Boolean network; Thomas' conjecture; Fixed point; Attractive cycle; Positive circuit; Interaction graph; Multistationarity; Boolean Jacobian matrix

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DOI: http://dx.doi.org/10.26713%2Fjims.v3i1.38

eISSN 0975-5748; pISSN 0974-875X