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We consider the following question on the relationship between the asymptoticbehaviours of asynchronous dynamics of Boolean networks and their regulatorystructures: does the presence of a cyclic attractor imply the existence of alocal negative circuit in the regulatory graph? When the number of modelcomponents $n$ verifies $n \geq 6$, the answer is known to be negative. We showthat the question can be translated into a Boolean satisfiability problem on $n\cdot 2^n$ variables. A Boolean formula expressing the absence of localnegative circuits and a necessary condition for the existence of cyclicattractors is found unsatisfiable for $n \leq 5$. In other words, for Booleannetworks with up to $5$ components, the presence of a cyclic attractor requiresthe existence of a local negative circuit.

Local negative circuits and cyclic attractors in Boolean networks with at most five components