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This paper presents a novel recurrent fuzzy neural network, called an interactively recurrent self-evolving fuzzy neural network (IRSFNN), for prediction and identification of dynamic systems. The recurrent structure in an IRSFNN is formed as an external loops and internal feedback by feeding the rule firing strength of each rule to others rules and itself. The consequent part in the IRSFNN is composed of a Takagi-Sugeno-Kang (TSK) or functional-link-based type. The proposed IRSFNN employs a functional link neural network (FLNN) to the consequent part of fuzzy rules for promoting the mapping ability. Unlike a TSK-type fuzzy neural network, the FLNN in the consequent part is a nonlinear function of input variables. An IRSFNNs learning starts with an empty rule base and all of the rules are generated and learned online through a simultaneous structure and parameter learning. An on-line clustering algorithm is effective in generating fuzzy rules. The consequent update parameters are derived by a variable-dimensional Kalman filter algorithm. The premise and recurrent parameters are learned through a gradient descent algorithm. We test the IRSFNN for the prediction and identification of dynamic plants and compare it to other well-known recurrent FNNs. The proposed model obtains enhanced performance results.