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This paper presents a new approach to the stability analysis of recurrent fuzzy systems (RFSs). RFSs are rule-based dynamic fuzzy systems that are usually obtained from heuristic or data-driven modeling. In the presented approach, the stability of both continuous-time and discrete-time RFS can be analyzed in a common framework. It is based on the representation of an RFS as a hybrid polynomial system. Due to the polynomial structure, the recently developed method of sum-of-squares (SOS) decomposition along with semidefinite programming can be employed to derive sufficient conditions for the stability of equilibrium points. We consider stability analysis for known equilibrium points as well as unknown equilibrium points. The latter results in a two-step procedure. In the first step, a polynomial is constructed. In the second verification step, this polynomial is proven to be a Lyapunov function for the RFS. The applicability of the approach is shown by two systems formulated as a rule-based RFS.