Being inspired by the Hopfield neural networks (Hopfield (1982) and Hopfield and Tank (1985)) and the nonlinear sigmoid power control algorithm for cellular radio systems in Uykan and Koivo (2004), in this paper, we present a novel discrete recurrent nonlinear system and extend the results in Uykan (2009), which are for autonomous linear systems, to nonlinear case. The proposed system can be viewed as a discrete-time realization of a recently proposed continuous-time network in Uykan (2013). In this paper, we focus on discrete-time analysis and provide various novel key results concerning the discrete-time dynamics of the proposed system, some of which are as follows: (i) the proposed system is shown to be stable in synchronous and asynchronous work mode in discrete time; (ii) a novel concept called Pseudo-SINR (pseudo-signal-to-interference-noise ratio) is introduced for discrete-time nonlinear systems; (iii) it is shown that when the system states approach an equilibrium point, the instantaneous Pseudo-SINRs are balanced; that is, they are equal to a target value. The simulation results confirm the novel results presented and show the effectiveness of the proposed discrete-time network as applied to various associative memory systems and clustering problems.