Ultimate “SIR” in Autonomous Linear Networks with Symmetric Weight Matrices, and Its Use to Stabilize the Network - A Hopfield-like network

In this paper, we present and analyse two Hopfield-like nonlinear networks, in continuous-time and discrete-time respectively. The proposed network is based on an autonomous linear system with a symmetric weight matrix, which is designed to be unstable, and a nonlinear function stabilizing the whole network thanks to a manipulated state variable called“ultimate SIR”. This variable is observed to be equal to the traditional Signal-to-Interference Ratio (SIR) definition in telecommunications engineering. The underlying linear system of the proposed continuous-time network is ẋ = Bx where B is a real symmetric matrix whose diagonal elements are fixed to a constant. The nonlinear function, on the other hand, is based on the defined system variables called “SIR”s. We also show that the “SIR”s of all the states converge to a constant value, called “system-specific Ultimate SIR”; which is equal to